{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0001","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-12)^0 的精確值。","answer":{"kind":"numeric","value":"1","explanation_zh":"(-12)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031001,"parameters":{"a":-12,"n":0,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-12,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0002","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-11)^1 的精確值。","answer":{"kind":"numeric","value":"-11","explanation_zh":"(-11)^1=-11。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031002,"parameters":{"a":-11,"n":1,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-11,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0003","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-10)^2 的精確值。","answer":{"kind":"numeric","value":"100","explanation_zh":"(-10)^2=100。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031003,"parameters":{"a":-10,"n":2,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-10,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0004","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-9)^3 的精確值。","answer":{"kind":"numeric","value":"-729","explanation_zh":"(-9)^3=-729。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031004,"parameters":{"a":-9,"n":3,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-9,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0005","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-8)^4 的精確值。","answer":{"kind":"numeric","value":"4096","explanation_zh":"(-8)^4=4096。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031005,"parameters":{"a":-8,"n":4,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-8,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0006","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-7)^5 的精確值。","answer":{"kind":"numeric","value":"-16807","explanation_zh":"(-7)^5=-16807。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031006,"parameters":{"a":-7,"n":5,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-7,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0007","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-6)^0 的精確值。","answer":{"kind":"numeric","value":"1","explanation_zh":"(-6)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031007,"parameters":{"a":-6,"n":0,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-6,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0008","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-5)^1 的精確值。","answer":{"kind":"numeric","value":"-5","explanation_zh":"(-5)^1=-5。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031008,"parameters":{"a":-5,"n":1,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-5,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0009","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-4)^2 的精確值。","answer":{"kind":"numeric","value":"16","explanation_zh":"(-4)^2=16。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031009,"parameters":{"a":-4,"n":2,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-4,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0010","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-3)^3 的精確值。","answer":{"kind":"numeric","value":"-27","explanation_zh":"(-3)^3=-27。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031010,"parameters":{"a":-3,"n":3,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-3,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0011","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-2)^4 的精確值。","answer":{"kind":"numeric","value":"16","explanation_zh":"(-2)^4=16。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031011,"parameters":{"a":-2,"n":4,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-2,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0012","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (-1)^5 的精確值。","answer":{"kind":"numeric","value":"-1","explanation_zh":"(-1)^5=-1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031012,"parameters":{"a":-1,"n":5,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":-1,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0013","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (0)^0 的精確值。","answer":{"kind":"numeric","value":"1","explanation_zh":"(0)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031013,"parameters":{"a":0,"n":0,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":0,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0014","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (0)^1 的精確值。","answer":{"kind":"numeric","value":"0","explanation_zh":"(0)^1=0。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031014,"parameters":{"a":0,"n":1,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":0,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0015","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (2)^2 的精確值。","answer":{"kind":"numeric","value":"4","explanation_zh":"(2)^2=4。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031015,"parameters":{"a":2,"n":2,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":2,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0016","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (3)^3 的精確值。","answer":{"kind":"numeric","value":"27","explanation_zh":"(3)^3=27。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031016,"parameters":{"a":3,"n":3,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":3,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0017","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (4)^4 的精確值。","answer":{"kind":"numeric","value":"256","explanation_zh":"(4)^4=256。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031017,"parameters":{"a":4,"n":4,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":4,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0018","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (5)^5 的精確值。","answer":{"kind":"numeric","value":"3125","explanation_zh":"(5)^5=3125。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031018,"parameters":{"a":5,"n":5,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":5,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0019","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (6)^0 的精確值。","answer":{"kind":"numeric","value":"1","explanation_zh":"(6)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031019,"parameters":{"a":6,"n":0,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":6,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0020","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (7)^1 的精確值。","answer":{"kind":"numeric","value":"7","explanation_zh":"(7)^1=7。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031020,"parameters":{"a":7,"n":1,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":7,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0021","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (8)^2 的精確值。","answer":{"kind":"numeric","value":"64","explanation_zh":"(8)^2=64。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031021,"parameters":{"a":8,"n":2,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":8,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0022","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (9)^3 的精確值。","answer":{"kind":"numeric","value":"729","explanation_zh":"(9)^3=729。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031022,"parameters":{"a":9,"n":3,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":9,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0023","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (10)^4 的精確值。","answer":{"kind":"numeric","value":"10000","explanation_zh":"(10)^4=10000。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031023,"parameters":{"a":10,"n":4,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":10,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0024","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (11)^5 的精確值。","answer":{"kind":"numeric","value":"161051","explanation_zh":"(11)^5=161051。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031024,"parameters":{"a":11,"n":5,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":11,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0025","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的精確值","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），求 (12)^0 的精確值。","answer":{"kind":"numeric","value":"1","explanation_zh":"(12)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的精確值"]},"variant":{"family_id":"qf-natural-power-exact","seed":202610031025,"parameters":{"a":12,"n":0,"domain":"reals","zero_power":"empty_product","operation":"natural_power","semantic_template":"qf-natural-power-exact/{\"a\":12,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0026","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(0)^0 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(0)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031026,"parameters":{"a":0,"n":0,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":0,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0027","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-9)^1 的結果是正、負還是零？","answer":{"kind":"short_text","value":"負","explanation_zh":"(-9)^1=-9。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031027,"parameters":{"a":-9,"n":1,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-9,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0028","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-8)^2 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(-8)^2=64。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031028,"parameters":{"a":-8,"n":2,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-8,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0029","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-7)^3 的結果是正、負還是零？","answer":{"kind":"short_text","value":"負","explanation_zh":"(-7)^3=-343。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031029,"parameters":{"a":-7,"n":3,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-7,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0030","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-6)^4 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(-6)^4=1296。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031030,"parameters":{"a":-6,"n":4,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-6,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0031","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-5)^5 的結果是正、負還是零？","answer":{"kind":"short_text","value":"負","explanation_zh":"(-5)^5=-3125。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031031,"parameters":{"a":-5,"n":5,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-5,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0032","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-4)^6 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(-4)^6=4096。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031032,"parameters":{"a":-4,"n":6,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-4,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0033","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-3)^0 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(-3)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031033,"parameters":{"a":-3,"n":0,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-3,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0034","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-2)^1 的結果是正、負還是零？","answer":{"kind":"short_text","value":"負","explanation_zh":"(-2)^1=-2。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031034,"parameters":{"a":-2,"n":1,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-2,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0035","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(-1)^2 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(-1)^2=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031035,"parameters":{"a":-1,"n":2,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":-1,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0036","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(0)^3 的結果是正、負還是零？","answer":{"kind":"short_text","value":"零","explanation_zh":"(0)^3=0。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031036,"parameters":{"a":0,"n":3,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":0,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0037","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(1)^4 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(1)^4=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031037,"parameters":{"a":1,"n":4,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":1,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0038","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(2)^5 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(2)^5=32。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031038,"parameters":{"a":2,"n":5,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":2,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0039","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(3)^6 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(3)^6=729。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031039,"parameters":{"a":3,"n":6,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":3,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0040","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(4)^0 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(4)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031040,"parameters":{"a":4,"n":0,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":4,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0041","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(5)^1 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(5)^1=5。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031041,"parameters":{"a":5,"n":1,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":5,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0042","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(6)^2 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(6)^2=36。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031042,"parameters":{"a":6,"n":2,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":6,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0043","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(7)^3 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(7)^3=343。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031043,"parameters":{"a":7,"n":3,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":7,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0044","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(8)^4 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(8)^4=4096。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031044,"parameters":{"a":8,"n":4,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":8,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0045","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(9)^5 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(9)^5=59049。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031045,"parameters":{"a":9,"n":5,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":9,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0046","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(10)^6 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(10)^6=1000000。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031046,"parameters":{"a":10,"n":6,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":10,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0047","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(11)^0 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(11)^0=1。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031047,"parameters":{"a":11,"n":0,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":11,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0048","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(12)^1 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(12)^1=12。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031048,"parameters":{"a":12,"n":1,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":12,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0049","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(13)^2 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(13)^2=169。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031049,"parameters":{"a":13,"n":2,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":13,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0050","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"自然數冪的符號","stem_zh":"在實數底數與自然數指數下採空乘積 a^0=1（此代數約定包含 0^0=1），(14)^3 的結果是正、負還是零？","answer":{"kind":"short_text","value":"正","explanation_zh":"(14)^3=2744。指數零時用空乘積；正指數才由底數符號及奇偶性判斷。此約定不是極限規則。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自然數冪的符號"]},"variant":{"family_id":"qf-natural-power-sign","seed":202610031050,"parameters":{"a":14,"n":3,"domain":"reals","zero_power":"empty_product","operation":"power_sign","semantic_template":"qf-natural-power-sign/{\"a\":14,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0051","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √0 的精確非負值。","answer":{"kind":"numeric","value":"0","explanation_zh":"0²=0 且 0≥0，所以主平方根取 0，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031051,"parameters":{"r":0,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0052","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √1 的精確非負值。","answer":{"kind":"numeric","value":"1","explanation_zh":"1²=1 且 1≥0，所以主平方根取 1，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031052,"parameters":{"r":1,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0053","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √4 的精確非負值。","answer":{"kind":"numeric","value":"2","explanation_zh":"2²=4 且 2≥0，所以主平方根取 2，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031053,"parameters":{"r":4,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0054","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √9 的精確非負值。","answer":{"kind":"numeric","value":"3","explanation_zh":"3²=9 且 3≥0，所以主平方根取 3，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031054,"parameters":{"r":9,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0055","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √16 的精確非負值。","answer":{"kind":"numeric","value":"4","explanation_zh":"4²=16 且 4≥0，所以主平方根取 4，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031055,"parameters":{"r":16,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0056","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √25 的精確非負值。","answer":{"kind":"numeric","value":"5","explanation_zh":"5²=25 且 5≥0，所以主平方根取 5，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031056,"parameters":{"r":25,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0057","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √36 的精確非負值。","answer":{"kind":"numeric","value":"6","explanation_zh":"6²=36 且 6≥0，所以主平方根取 6，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031057,"parameters":{"r":36,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0058","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √49 的精確非負值。","answer":{"kind":"numeric","value":"7","explanation_zh":"7²=49 且 7≥0，所以主平方根取 7，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031058,"parameters":{"r":49,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":49}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0059","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √64 的精確非負值。","answer":{"kind":"numeric","value":"8","explanation_zh":"8²=64 且 8≥0，所以主平方根取 8，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031059,"parameters":{"r":64,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":64}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0060","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √81 的精確非負值。","answer":{"kind":"numeric","value":"9","explanation_zh":"9²=81 且 9≥0，所以主平方根取 9，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031060,"parameters":{"r":81,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":81}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0061","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √100 的精確非負值。","answer":{"kind":"numeric","value":"10","explanation_zh":"10²=100 且 10≥0，所以主平方根取 10，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031061,"parameters":{"r":100,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":100}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0062","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √121 的精確非負值。","answer":{"kind":"numeric","value":"11","explanation_zh":"11²=121 且 11≥0，所以主平方根取 11，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031062,"parameters":{"r":121,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":121}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0063","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √144 的精確非負值。","answer":{"kind":"numeric","value":"12","explanation_zh":"12²=144 且 12≥0，所以主平方根取 12，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031063,"parameters":{"r":144,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":144}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0064","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √169 的精確非負值。","answer":{"kind":"numeric","value":"13","explanation_zh":"13²=169 且 13≥0，所以主平方根取 13，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031064,"parameters":{"r":169,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":169}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0065","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √196 的精確非負值。","answer":{"kind":"numeric","value":"14","explanation_zh":"14²=196 且 14≥0，所以主平方根取 14，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031065,"parameters":{"r":196,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":196}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0066","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √225 的精確非負值。","answer":{"kind":"numeric","value":"15","explanation_zh":"15²=225 且 15≥0，所以主平方根取 15，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031066,"parameters":{"r":225,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":225}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0067","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √256 的精確非負值。","answer":{"kind":"numeric","value":"16","explanation_zh":"16²=256 且 16≥0，所以主平方根取 16，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031067,"parameters":{"r":256,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":256}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0068","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √289 的精確非負值。","answer":{"kind":"numeric","value":"17","explanation_zh":"17²=289 且 17≥0，所以主平方根取 17，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031068,"parameters":{"r":289,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":289}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0069","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √324 的精確非負值。","answer":{"kind":"numeric","value":"18","explanation_zh":"18²=324 且 18≥0，所以主平方根取 18，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031069,"parameters":{"r":324,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":324}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0070","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √361 的精確非負值。","answer":{"kind":"numeric","value":"19","explanation_zh":"19²=361 且 19≥0，所以主平方根取 19，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031070,"parameters":{"r":361,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":361}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0071","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √400 的精確非負值。","answer":{"kind":"numeric","value":"20","explanation_zh":"20²=400 且 20≥0，所以主平方根取 20，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031071,"parameters":{"r":400,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":400}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0072","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √441 的精確非負值。","answer":{"kind":"numeric","value":"21","explanation_zh":"21²=441 且 21≥0，所以主平方根取 21，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031072,"parameters":{"r":441,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":441}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0073","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √484 的精確非負值。","answer":{"kind":"numeric","value":"22","explanation_zh":"22²=484 且 22≥0，所以主平方根取 22，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031073,"parameters":{"r":484,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":484}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0074","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √529 的精確非負值。","answer":{"kind":"numeric","value":"23","explanation_zh":"23²=529 且 23≥0，所以主平方根取 23，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031074,"parameters":{"r":529,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":529}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0075","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"完全平方數的主平方根","stem_zh":"在非負實數上取主平方根，求 √576 的精確非負值。","answer":{"kind":"numeric","value":"24","explanation_zh":"24²=576 且 24≥0，所以主平方根取 24，不是正負兩個分支。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["完全平方數的主平方根"]},"variant":{"family_id":"qf-principal-square-root-exact","seed":202610031075,"parameters":{"r":576,"domain":"reals","operation":"principal_sqrt","semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":576}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0076","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-12 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-12，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031076,"parameters":{"r":-12,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0077","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-11 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-11，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031077,"parameters":{"r":-11,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0078","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-10 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-10，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031078,"parameters":{"r":-10,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0079","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-9 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-9，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031079,"parameters":{"r":-9,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0080","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-8 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-8，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031080,"parameters":{"r":-8,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0081","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-7 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-7，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031081,"parameters":{"r":-7,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0082","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-6 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-6，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031082,"parameters":{"r":-6,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0083","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-5 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-5，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031083,"parameters":{"r":-5,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0084","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-4 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-4，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031084,"parameters":{"r":-4,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0085","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-3 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-3，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031085,"parameters":{"r":-3,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0086","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-2 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-2，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031086,"parameters":{"r":-2,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0087","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=-1 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"否","explanation_zh":"r=-1，違反 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031087,"parameters":{"r":-1,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0088","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=0 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=0，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031088,"parameters":{"r":0,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0089","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=1 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=1，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031089,"parameters":{"r":1,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0091","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=3 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=3，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031091,"parameters":{"r":3,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0092","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=4 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=4，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031092,"parameters":{"r":4,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0093","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=5 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=5，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031093,"parameters":{"r":5,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0094","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=6 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=6，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031094,"parameters":{"r":6,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0095","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=7 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=7，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031095,"parameters":{"r":7,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0096","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=8 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=8，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031096,"parameters":{"r":8,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0097","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=9 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=9，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031097,"parameters":{"r":9,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0098","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=10 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=10，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031098,"parameters":{"r":10,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0099","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=11 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=11，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031099,"parameters":{"r":11,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r01-power-root-0100","batch_id":"batch-20261003-r01-power-root-foundations","language":"zh-Hant","source_mko_ids":["mko-power-root"],"title_zh":"主平方根的實數定義域","stem_zh":"本題實數主平方根只對非負輸入定義。r=12 是否在主平方根定義域內？","answer":{"kind":"short_text","value":"是","explanation_zh":"r=12，符合 r≥0。負輸入的補零程式慣例不屬於這個主平方根定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["主平方根的實數定義域"]},"variant":{"family_id":"qf-principal-square-root-domain","seed":202610031100,"parameters":{"r":12,"domain":"reals","operation":"sqrt_domain","semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}