{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0001","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"864eadb6cdb4d5eefdf011fc61534e543e0331dd36b6ae2c4c5369e7b583741b","family_id":"qf-natural-power-exact","seed":202610031001,"semantic_template":"qf-natural-power-exact/{\"a\":-12,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-12)^0 exactly.","answer":{"kind":"numeric","value":"1","explanation":"(-12)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0002","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d168fd9da5959a389132c737195533a2b67e73517078a8e6c728be302c40dfac","family_id":"qf-natural-power-exact","seed":202610031002,"semantic_template":"qf-natural-power-exact/{\"a\":-11,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-11)^1 exactly.","answer":{"kind":"numeric","value":"-11","explanation":"(-11)^1=-11. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0003","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ab5d0ba4bfb704e2f255b197253d4e7b60d41b32f4acccca8183080727faaa3","family_id":"qf-natural-power-exact","seed":202610031003,"semantic_template":"qf-natural-power-exact/{\"a\":-10,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-10)^2 exactly.","answer":{"kind":"numeric","value":"100","explanation":"(-10)^2=100. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0004","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02a2fd30dd88919e82c8cc6f160d4b59eaff9d76e8469370f932feb88a44f3df","family_id":"qf-natural-power-exact","seed":202610031004,"semantic_template":"qf-natural-power-exact/{\"a\":-9,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-9)^3 exactly.","answer":{"kind":"numeric","value":"-729","explanation":"(-9)^3=-729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0005","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090aedef67fce9acdb7659fdbb74a809c1c931f91525d9ed1d986a257e6a0c86","family_id":"qf-natural-power-exact","seed":202610031005,"semantic_template":"qf-natural-power-exact/{\"a\":-8,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-8)^4 exactly.","answer":{"kind":"numeric","value":"4096","explanation":"(-8)^4=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0006","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6ee486aec4c90a81263a0e228bf6ec6c39b732436d4601438743dc19a356869","family_id":"qf-natural-power-exact","seed":202610031006,"semantic_template":"qf-natural-power-exact/{\"a\":-7,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-7)^5 exactly.","answer":{"kind":"numeric","value":"-16807","explanation":"(-7)^5=-16807. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0007","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"155c0c503fad11d75c87f90aed9477f3dd18e206ff73ea1c7d159be58876d35d","family_id":"qf-natural-power-exact","seed":202610031007,"semantic_template":"qf-natural-power-exact/{\"a\":-6,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-6)^0 exactly.","answer":{"kind":"numeric","value":"1","explanation":"(-6)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0008","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b013c2fd030fe7ac4b4d87bd32bb9ab62fabb13c51cca7411d0f2a2f7d60ed6","family_id":"qf-natural-power-exact","seed":202610031008,"semantic_template":"qf-natural-power-exact/{\"a\":-5,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-5)^1 exactly.","answer":{"kind":"numeric","value":"-5","explanation":"(-5)^1=-5. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0009","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e992b342a2023af2918694a89806e10593e5bbf1db671f071b614b68acb0c123","family_id":"qf-natural-power-exact","seed":202610031009,"semantic_template":"qf-natural-power-exact/{\"a\":-4,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-4)^2 exactly.","answer":{"kind":"numeric","value":"16","explanation":"(-4)^2=16. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0010","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d761df3b00c2bf719c63de96a78cb98f5751c6c4f1302c815d092f905b6867ec","family_id":"qf-natural-power-exact","seed":202610031010,"semantic_template":"qf-natural-power-exact/{\"a\":-3,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-3)^3 exactly.","answer":{"kind":"numeric","value":"-27","explanation":"(-3)^3=-27. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0011","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9df56c956f19ab8bff3517578d7179fc3bd43667b9b68c35f2e9d208adc7556","family_id":"qf-natural-power-exact","seed":202610031011,"semantic_template":"qf-natural-power-exact/{\"a\":-2,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-2)^4 exactly.","answer":{"kind":"numeric","value":"16","explanation":"(-2)^4=16. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0012","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40bff7535cc968fe46cac3c745147e51c6388f11a1987c6a68be483cec2e539f","family_id":"qf-natural-power-exact","seed":202610031012,"semantic_template":"qf-natural-power-exact/{\"a\":-1,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-1)^5 exactly.","answer":{"kind":"numeric","value":"-1","explanation":"(-1)^5=-1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0013","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98fe00066b45dd671aea9d215d8e03859213b5e00ea28022d7720cf76b249ac4","family_id":"qf-natural-power-exact","seed":202610031013,"semantic_template":"qf-natural-power-exact/{\"a\":0,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (0)^0 exactly.","answer":{"kind":"numeric","value":"1","explanation":"(0)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0014","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4c7e860e9cca2eb6549ee6872f60ef29ceea0ec7b86a8f5352202e41b5f791c","family_id":"qf-natural-power-exact","seed":202610031014,"semantic_template":"qf-natural-power-exact/{\"a\":0,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (0)^1 exactly.","answer":{"kind":"numeric","value":"0","explanation":"(0)^1=0. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0015","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05e299e45b03fdd26cddbf8a4544afbf1d8bfcddf5dcba04a4aa55e4d5ecd04a","family_id":"qf-natural-power-exact","seed":202610031015,"semantic_template":"qf-natural-power-exact/{\"a\":2,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (2)^2 exactly.","answer":{"kind":"numeric","value":"4","explanation":"(2)^2=4. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0016","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48061d7580fdb4777aad98bdb003f60279b252b0de39d203deb5db34f9dda72e","family_id":"qf-natural-power-exact","seed":202610031016,"semantic_template":"qf-natural-power-exact/{\"a\":3,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (3)^3 exactly.","answer":{"kind":"numeric","value":"27","explanation":"(3)^3=27. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0017","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce049e7425ea3a4450f8628cb37ea44418cfea764e1573eedf663e8e74a5c393","family_id":"qf-natural-power-exact","seed":202610031017,"semantic_template":"qf-natural-power-exact/{\"a\":4,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (4)^4 exactly.","answer":{"kind":"numeric","value":"256","explanation":"(4)^4=256. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0018","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb506b07221d09702b33350fce2c3d07269bac72cf66a0de2dd6d8b7e3a5b938","family_id":"qf-natural-power-exact","seed":202610031018,"semantic_template":"qf-natural-power-exact/{\"a\":5,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (5)^5 exactly.","answer":{"kind":"numeric","value":"3125","explanation":"(5)^5=3125. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0019","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"282bc50682d071d093fe5356de4e2add2da9fad2687bd9ffbf013d29858ff149","family_id":"qf-natural-power-exact","seed":202610031019,"semantic_template":"qf-natural-power-exact/{\"a\":6,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (6)^0 exactly.","answer":{"kind":"numeric","value":"1","explanation":"(6)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0020","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf80d4a8cd11936fc4979f609597aad64cc17f77fc4b8ec8d46213f37b581d47","family_id":"qf-natural-power-exact","seed":202610031020,"semantic_template":"qf-natural-power-exact/{\"a\":7,\"domain\":\"reals\",\"n\":1,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (7)^1 exactly.","answer":{"kind":"numeric","value":"7","explanation":"(7)^1=7. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0021","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef0888370c8125681197d479f30afe22932f3ac5cf671003da7f11d9ff20abd5","family_id":"qf-natural-power-exact","seed":202610031021,"semantic_template":"qf-natural-power-exact/{\"a\":8,\"domain\":\"reals\",\"n\":2,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (8)^2 exactly.","answer":{"kind":"numeric","value":"64","explanation":"(8)^2=64. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0022","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd30ab82bd84e468616ba1d6e8dd2a80a08bb6525819ce102fd0b4f77a35e271","family_id":"qf-natural-power-exact","seed":202610031022,"semantic_template":"qf-natural-power-exact/{\"a\":9,\"domain\":\"reals\",\"n\":3,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (9)^3 exactly.","answer":{"kind":"numeric","value":"729","explanation":"(9)^3=729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0023","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cde8fdd54e86b6e60ea0280ba10c3aec5d27c5c124fe1400ef69986bb8b8c4c4","family_id":"qf-natural-power-exact","seed":202610031023,"semantic_template":"qf-natural-power-exact/{\"a\":10,\"domain\":\"reals\",\"n\":4,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (10)^4 exactly.","answer":{"kind":"numeric","value":"10000","explanation":"(10)^4=10000. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0024","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ab541d171b560ae001bd1110ade096de012c2fa95faee0a135677d18c8fbbe","family_id":"qf-natural-power-exact","seed":202610031024,"semantic_template":"qf-natural-power-exact/{\"a\":11,\"domain\":\"reals\",\"n\":5,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (11)^5 exactly.","answer":{"kind":"numeric","value":"161051","explanation":"(11)^5=161051. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0025","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56ea32f4671e149019ed27263517f78eb576d72e79bfe9b7142624dc7bd04cd7","family_id":"qf-natural-power-exact","seed":202610031025,"semantic_template":"qf-natural-power-exact/{\"a\":12,\"domain\":\"reals\",\"n\":0,\"operation\":\"natural_power\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Exact Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (12)^0 exactly.","answer":{"kind":"numeric","value":"1","explanation":"(12)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Exact Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0026","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7cd405bd64a7be486d7c9c0add63dd99598fdb97eac0d0abfbf0a1e235b3d98","family_id":"qf-natural-power-sign","seed":202610031026,"semantic_template":"qf-natural-power-sign/{\"a\":0,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (0)^0 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(0)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0027","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9e658e1f967ecab6c93e4c9e17858167200b6a8540b52cc76ad8831bf907c4c","family_id":"qf-natural-power-sign","seed":202610031027,"semantic_template":"qf-natural-power-sign/{\"a\":-9,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-9)^1 positive, negative, or zero?","answer":{"kind":"short_text","value":"negative","explanation":"(-9)^1=-9. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0028","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"346c6b96320336cc282bbbf1e0cce13a41ade8797b2372feba9fb330141574cc","family_id":"qf-natural-power-sign","seed":202610031028,"semantic_template":"qf-natural-power-sign/{\"a\":-8,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-8)^2 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(-8)^2=64. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0029","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f510bc1352f5f021219b4f6480ed4d47935e492663efef64a859ccca05c03236","family_id":"qf-natural-power-sign","seed":202610031029,"semantic_template":"qf-natural-power-sign/{\"a\":-7,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-7)^3 positive, negative, or zero?","answer":{"kind":"short_text","value":"negative","explanation":"(-7)^3=-343. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0030","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f3d99d7bf2372a25bbac4c4bbf1210dc928672d478f0f35be0838c07778f71","family_id":"qf-natural-power-sign","seed":202610031030,"semantic_template":"qf-natural-power-sign/{\"a\":-6,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-6)^4 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(-6)^4=1296. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0031","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d15b1965632cbe88e5717324e5817b7429d9557f82ef296c88bc6de12d5359c4","family_id":"qf-natural-power-sign","seed":202610031031,"semantic_template":"qf-natural-power-sign/{\"a\":-5,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-5)^5 positive, negative, or zero?","answer":{"kind":"short_text","value":"negative","explanation":"(-5)^5=-3125. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0032","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dfb5ed3db2442f4e3dfca034730dbb6c4fe76f0146b924b3043a2a8019ab403","family_id":"qf-natural-power-sign","seed":202610031032,"semantic_template":"qf-natural-power-sign/{\"a\":-4,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-4)^6 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(-4)^6=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0033","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86fd3867871009656b009f4856258ada4c8ca9850e9a140bc07fec3031b3535e","family_id":"qf-natural-power-sign","seed":202610031033,"semantic_template":"qf-natural-power-sign/{\"a\":-3,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-3)^0 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(-3)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0034","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"086364737588c762d3c54c0518c5441fba9c13f93983f8e42db842614b47e30a","family_id":"qf-natural-power-sign","seed":202610031034,"semantic_template":"qf-natural-power-sign/{\"a\":-2,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-2)^1 positive, negative, or zero?","answer":{"kind":"short_text","value":"negative","explanation":"(-2)^1=-2. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0035","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b637cf267c1119b5dc33f42a540e653f7a6a9152ee24774f9c3a96ac708ae0","family_id":"qf-natural-power-sign","seed":202610031035,"semantic_template":"qf-natural-power-sign/{\"a\":-1,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-1)^2 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(-1)^2=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0036","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e297d102dc078ed61b83fbbc0b690782c3a079b7f6e897577abd3d9383be1c2e","family_id":"qf-natural-power-sign","seed":202610031036,"semantic_template":"qf-natural-power-sign/{\"a\":0,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (0)^3 positive, negative, or zero?","answer":{"kind":"short_text","value":"zero","explanation":"(0)^3=0. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0037","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"171ba754dbf39a46e77a7e406ab81db143728421954ee85df9dca95a9cb75e1a","family_id":"qf-natural-power-sign","seed":202610031037,"semantic_template":"qf-natural-power-sign/{\"a\":1,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (1)^4 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(1)^4=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0038","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f9397051d88d47ff7a2483e84053d33b332e771047011d356d158f31019431e","family_id":"qf-natural-power-sign","seed":202610031038,"semantic_template":"qf-natural-power-sign/{\"a\":2,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (2)^5 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(2)^5=32. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0039","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0f1555edaf0b0a233e9f0dcd3dd04aabd2efa881b1a783cb50a1a916c7a3c65","family_id":"qf-natural-power-sign","seed":202610031039,"semantic_template":"qf-natural-power-sign/{\"a\":3,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (3)^6 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(3)^6=729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0040","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56ffedf0c4b1cb5cec63f6367efe10ce09f097af7a67570fd2e05009f8269f0d","family_id":"qf-natural-power-sign","seed":202610031040,"semantic_template":"qf-natural-power-sign/{\"a\":4,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (4)^0 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(4)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0041","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca8ace95a096f2602ea1b6f0b249f18eb7dc038febe756be0b6f6d7559fa2639","family_id":"qf-natural-power-sign","seed":202610031041,"semantic_template":"qf-natural-power-sign/{\"a\":5,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (5)^1 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(5)^1=5. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0042","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"817fdad7404cb7d16b05215fd35b21dbe1d2ff3d2e8fc5de848e32d29ea84954","family_id":"qf-natural-power-sign","seed":202610031042,"semantic_template":"qf-natural-power-sign/{\"a\":6,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (6)^2 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(6)^2=36. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0043","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a1e67cd51a3ad7368f953bbab6e4bf0391170d72a24107f6833f00a99daa786","family_id":"qf-natural-power-sign","seed":202610031043,"semantic_template":"qf-natural-power-sign/{\"a\":7,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (7)^3 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(7)^3=343. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0044","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb505faaf103815147ea655ccc48733180495f683cb78bafd3615c6f9902af7","family_id":"qf-natural-power-sign","seed":202610031044,"semantic_template":"qf-natural-power-sign/{\"a\":8,\"domain\":\"reals\",\"n\":4,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (8)^4 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(8)^4=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0045","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a67aae4f0088f37a23995a20b2c5f5a6ca424ff86af6ddc68ccc8303050989e","family_id":"qf-natural-power-sign","seed":202610031045,"semantic_template":"qf-natural-power-sign/{\"a\":9,\"domain\":\"reals\",\"n\":5,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (9)^5 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(9)^5=59049. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0046","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b516b576e1057d954eb2c47461c71f9ded909828928a491fab823d5093b015d4","family_id":"qf-natural-power-sign","seed":202610031046,"semantic_template":"qf-natural-power-sign/{\"a\":10,\"domain\":\"reals\",\"n\":6,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (10)^6 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(10)^6=1000000. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0047","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"166453e2eb4334dd80dd53b861c629343b12db689a129038ce2b83e9536cb070","family_id":"qf-natural-power-sign","seed":202610031047,"semantic_template":"qf-natural-power-sign/{\"a\":11,\"domain\":\"reals\",\"n\":0,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (11)^0 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(11)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0048","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33f5503361fe5631ae636c2cc8769e555c712b27232059ddd4e3584f016d3b4a","family_id":"qf-natural-power-sign","seed":202610031048,"semantic_template":"qf-natural-power-sign/{\"a\":12,\"domain\":\"reals\",\"n\":1,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (12)^1 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(12)^1=12. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0049","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c9327162fe9abb6a1535018f1c9da75a3442df4f7a879b59613efea7635fec","family_id":"qf-natural-power-sign","seed":202610031049,"semantic_template":"qf-natural-power-sign/{\"a\":13,\"domain\":\"reals\",\"n\":2,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (13)^2 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(13)^2=169. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0050","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f55aaf2ba3ba7c31db7b41ef64eac5f21e089fc282771bd75083d811d374648","family_id":"qf-natural-power-sign","seed":202610031050,"semantic_template":"qf-natural-power-sign/{\"a\":14,\"domain\":\"reals\",\"n\":3,\"operation\":\"power_sign\",\"zero_power\":\"empty_product\"}"},"fields":{"title":"Sign of a Natural-Number Power","stem":"For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (14)^3 positive, negative, or zero?","answer":{"kind":"short_text","value":"positive","explanation":"(14)^3=2744. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits."},"skills":["Sign of a Natural-Number Power"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0051","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a7c9d0993b9a5facc5a34dea9efd1b351f4565d094a1a6f8d80fa1369baefb1","family_id":"qf-principal-square-root-exact","seed":202610031051,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":0}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √0.","answer":{"kind":"numeric","value":"0","explanation":"0²=0 and 0≥0, so the principal square root is 0; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0052","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"640e9198a544bbb6396db32066c8eeab28fa43525d369fcc2407b0c291a9d65d","family_id":"qf-principal-square-root-exact","seed":202610031052,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":1}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.","answer":{"kind":"numeric","value":"1","explanation":"1²=1 and 1≥0, so the principal square root is 1; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0053","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0c5464bb508735fe1c6209c7af74b44fa841b0c5bcc8978f0e4479518119c0","family_id":"qf-principal-square-root-exact","seed":202610031053,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":4}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √4.","answer":{"kind":"numeric","value":"2","explanation":"2²=4 and 2≥0, so the principal square root is 2; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0054","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b5f48695abedfc8ebec0658f00ca86eea16df8489373df26056432798884d5","family_id":"qf-principal-square-root-exact","seed":202610031054,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":9}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √9.","answer":{"kind":"numeric","value":"3","explanation":"3²=9 and 3≥0, so the principal square root is 3; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0055","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c2dbbd629304e383577fa9f32cb6a39b6a36d502124d96dfed5d71cc9c9de22","family_id":"qf-principal-square-root-exact","seed":202610031055,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":16}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √16.","answer":{"kind":"numeric","value":"4","explanation":"4²=16 and 4≥0, so the principal square root is 4; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0056","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73a3b7489acbf68dd0a0d2eacad835bccc687b093b5f58fbabbe7e66d994f948","family_id":"qf-principal-square-root-exact","seed":202610031056,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":25}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √25.","answer":{"kind":"numeric","value":"5","explanation":"5²=25 and 5≥0, so the principal square root is 5; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0057","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d43080ff6541237e805dfcff9f7246c898cde6e9e353caa95797c7694e04dccb","family_id":"qf-principal-square-root-exact","seed":202610031057,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":36}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √36.","answer":{"kind":"numeric","value":"6","explanation":"6²=36 and 6≥0, so the principal square root is 6; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0058","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af414f977ce8067afd54f3ee20913766db3108d4b7e77cf738dab2c18d7a48b","family_id":"qf-principal-square-root-exact","seed":202610031058,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":49}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √49.","answer":{"kind":"numeric","value":"7","explanation":"7²=49 and 7≥0, so the principal square root is 7; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0059","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8b254718508267b9ea54e9bc6148be43e40da2df87044a3d53a55f6a75b0bb4","family_id":"qf-principal-square-root-exact","seed":202610031059,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":64}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √64.","answer":{"kind":"numeric","value":"8","explanation":"8²=64 and 8≥0, so the principal square root is 8; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0060","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35157f6c17b5fad738fca976021839ccfee9c9791f682c37366105d0735b848e","family_id":"qf-principal-square-root-exact","seed":202610031060,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":81}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √81.","answer":{"kind":"numeric","value":"9","explanation":"9²=81 and 9≥0, so the principal square root is 9; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0061","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f10d05bf48573bca04a502280cecad430b8c967357967aef2ea14cd480be9cf2","family_id":"qf-principal-square-root-exact","seed":202610031061,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":100}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √100.","answer":{"kind":"numeric","value":"10","explanation":"10²=100 and 10≥0, so the principal square root is 10; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0062","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2695319392257f2a015b4dcd73e4190f504d7ae51d0b111b004385cf443343","family_id":"qf-principal-square-root-exact","seed":202610031062,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":121}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √121.","answer":{"kind":"numeric","value":"11","explanation":"11²=121 and 11≥0, so the principal square root is 11; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0063","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9cc1b8cb2792364c20c6e3073c38aa9082ea86b11d8a227d80721842b89e176","family_id":"qf-principal-square-root-exact","seed":202610031063,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":144}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √144.","answer":{"kind":"numeric","value":"12","explanation":"12²=144 and 12≥0, so the principal square root is 12; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0064","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ebafb214c1e9afcbe1cbb6d855564e0cefe9b2396c9704423a732130d190f5a","family_id":"qf-principal-square-root-exact","seed":202610031064,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":169}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √169.","answer":{"kind":"numeric","value":"13","explanation":"13²=169 and 13≥0, so the principal square root is 13; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0065","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7620fb1d93bc08f32759be5bc0aa29180668cf126a750afe7cb3333bbb9955f6","family_id":"qf-principal-square-root-exact","seed":202610031065,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":196}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √196.","answer":{"kind":"numeric","value":"14","explanation":"14²=196 and 14≥0, so the principal square root is 14; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0066","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40539c51da1fb35d63eb3acfcb0f1a9a373dd53a8d52b36a1d90642066a5ee3a","family_id":"qf-principal-square-root-exact","seed":202610031066,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":225}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √225.","answer":{"kind":"numeric","value":"15","explanation":"15²=225 and 15≥0, so the principal square root is 15; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0067","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52090907340f22ce79edcc0df9556f414ad6dbb00bf93ed49ea797ff96b6e8d6","family_id":"qf-principal-square-root-exact","seed":202610031067,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":256}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √256.","answer":{"kind":"numeric","value":"16","explanation":"16²=256 and 16≥0, so the principal square root is 16; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0068","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7efd87ae10cb12b39cf8b4bc794951d01c98fb901cb14c4b61d9cc6e3c80fc20","family_id":"qf-principal-square-root-exact","seed":202610031068,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":289}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √289.","answer":{"kind":"numeric","value":"17","explanation":"17²=289 and 17≥0, so the principal square root is 17; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0069","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03249b9d2e4cfdb7d7307ffd7927c0ebb92866ace91c78e1ea7741bc93b468f9","family_id":"qf-principal-square-root-exact","seed":202610031069,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":324}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √324.","answer":{"kind":"numeric","value":"18","explanation":"18²=324 and 18≥0, so the principal square root is 18; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0070","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa15d6714e9f1259ef9b3035d9519538c8eba01cf384f131c5f183432e91509f","family_id":"qf-principal-square-root-exact","seed":202610031070,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":361}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √361.","answer":{"kind":"numeric","value":"19","explanation":"19²=361 and 19≥0, so the principal square root is 19; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0071","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ebcddc0634995eb53ae6ed664fa550292427bc895dfbb76bf90215db1ac7d23","family_id":"qf-principal-square-root-exact","seed":202610031071,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":400}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √400.","answer":{"kind":"numeric","value":"20","explanation":"20²=400 and 20≥0, so the principal square root is 20; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0072","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1975a8e855155e62478f7e6e96975305bfd57c232c22ead4b2bbcc94dad5684f","family_id":"qf-principal-square-root-exact","seed":202610031072,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":441}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √441.","answer":{"kind":"numeric","value":"21","explanation":"21²=441 and 21≥0, so the principal square root is 21; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0073","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34197c446da3844ac4fdc5b1ef2de17c1844d094a6e37b11cf8c5923096e4c94","family_id":"qf-principal-square-root-exact","seed":202610031073,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":484}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √484.","answer":{"kind":"numeric","value":"22","explanation":"22²=484 and 22≥0, so the principal square root is 22; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0074","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76e154bde2665fbcd8ea4d25fd712d1b703db65b57a2a867085883394b8a4bbc","family_id":"qf-principal-square-root-exact","seed":202610031074,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":529}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √529.","answer":{"kind":"numeric","value":"23","explanation":"23²=529 and 23≥0, so the principal square root is 23; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0075","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da5e26cd386d6c08b3f748d531591bfd9a6533fef6d457fb8f2584cac200f956","family_id":"qf-principal-square-root-exact","seed":202610031075,"semantic_template":"qf-principal-square-root-exact/{\"domain\":\"reals\",\"operation\":\"principal_sqrt\",\"r\":576}"},"fields":{"title":"Principal Square Root of a Perfect Square","stem":"Using the principal square root on the nonnegative reals, find the exact nonnegative value of √576.","answer":{"kind":"numeric","value":"24","explanation":"24²=576 and 24≥0, so the principal square root is 24; it is a single value rather than two signed branches."},"skills":["Principal Square Root of a Perfect Square"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0076","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2bcd1f16c1aad04753adae1f9a91301f8a1cb523429d47253ec45ea51a3487a","family_id":"qf-principal-square-root-domain","seed":202610031076,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-12}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-12 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-12 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0077","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89c925a409437c66d5ce30dc4442556c5913b1485968f3a9635a1af4df49b777","family_id":"qf-principal-square-root-domain","seed":202610031077,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-11}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-11 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-11 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0078","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"161d481b2f61bf31c0f4a12da299612f7454c1c1ea76f00c60355e334769c8d1","family_id":"qf-principal-square-root-domain","seed":202610031078,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-10}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-10 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-10 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0079","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"852abeca8ee119b80d4047c3c0068961fe3342ff384f1b2b54a517d4e9b687c0","family_id":"qf-principal-square-root-domain","seed":202610031079,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-9}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-9 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-9 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0080","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f764b9ce2639855d5c816323ceed5963a010fc3b9a03b94aa35361f3c58416af","family_id":"qf-principal-square-root-domain","seed":202610031080,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-8}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-8 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-8 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0081","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d45025ac188321e24542f314d308f6cbf1244d12698da8bd2048f032bf2cac36","family_id":"qf-principal-square-root-domain","seed":202610031081,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-7}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-7 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-7 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0082","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b25561385f296d0827de8459865df5d5560f30e09f284aa9836606ba1929978d","family_id":"qf-principal-square-root-domain","seed":202610031082,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-6}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-6 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-6 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0083","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56d4a43a0b042777d25270384bfc892e9404010a0c5ed4311a9966ab3a738b79","family_id":"qf-principal-square-root-domain","seed":202610031083,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-5}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-5 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-5 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0084","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c723e21e07b1c25ecc02cf947c8eab0a1ee3f47acd56a06efffb6353b76e83c","family_id":"qf-principal-square-root-domain","seed":202610031084,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-4}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-4 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-4 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0085","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8c1278220b57c7232307ec30c230e2db66ebac2c89560ebd48e9f17d84c5ec8","family_id":"qf-principal-square-root-domain","seed":202610031085,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-3}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-3 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-3 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0086","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7789bff9575826b9ecdf00d0c84f2ae68bf9c89228c12b074857679734e6ce03","family_id":"qf-principal-square-root-domain","seed":202610031086,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-2}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-2 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-2 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0087","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"330bc303ad941cdfd8ebe851f86cacd04bfca7448e69397f781fbbb2f07f36e1","family_id":"qf-principal-square-root-domain","seed":202610031087,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":-1}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-1 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"r=-1 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0088","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e299b2c83bb197cbebc2ad3ec74ac02719d0352812769968c07087821a908b6","family_id":"qf-principal-square-root-domain","seed":202610031088,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":0}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=0 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=0 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0089","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf0c11d25c57ba339670f9b75a4efdd502da1f3bc6833d3c6fe6d8ccd2842a1","family_id":"qf-principal-square-root-domain","seed":202610031089,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":1}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=1 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=1 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0090","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4d2f0ea3634d3c6a1581939976d5a04fbf4e865351b3128f887ad701838e424","family_id":"qf-principal-square-root-domain","seed":202610031090,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":2}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=2 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=2 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0091","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a902375a27066afe564dfda1b47c1840bc2154f85b15f82a1985a03098dee86b","family_id":"qf-principal-square-root-domain","seed":202610031091,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":3}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=3 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=3 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0092","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"413fa50bb2851d65e0dccc2f5d256794d631093ff49abf5712213152f08f4f63","family_id":"qf-principal-square-root-domain","seed":202610031092,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":4}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=4 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=4 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0093","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee272263b0f734985841c37b0ab24c00e29dfca6b553cf474f435e7eb86e83ac","family_id":"qf-principal-square-root-domain","seed":202610031093,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":5}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=5 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=5 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0094","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c93326b60a4159ff2ecff3c3917dd9a7b8eb373e6066afdab4e8a28fd42c834","family_id":"qf-principal-square-root-domain","seed":202610031094,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":6}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=6 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=6 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0095","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f358e8240364f8b89eb142eaab4e36512097d78b445e712f386336959f44cb27","family_id":"qf-principal-square-root-domain","seed":202610031095,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":7}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=7 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=7 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0096","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62745ebf5f064cbaea13b10fdde3295a96e40e6d0a55c3e679f79de7143e4338","family_id":"qf-principal-square-root-domain","seed":202610031096,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":8}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=8 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=8 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0097","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d50a59d4678fe5a7d4c2338ede603c8b480aef4ef6ceec9daea142560ad1f6c4","family_id":"qf-principal-square-root-domain","seed":202610031097,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":9}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=9 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=9 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0098","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4241bc8d149e2c988172ece7de2ba1ed5ec76474486801bf3cfe174241f342f8","family_id":"qf-principal-square-root-domain","seed":202610031098,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":10}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=10 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=10 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0099","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"216a996e70b3345f862137a1caf25684110bbefce6a6f431fc00a984940738e5","family_id":"qf-principal-square-root-domain","seed":202610031099,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":11}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=11 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=11 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r01-power-root-0100","batch_id":"batch-20261003-r01-power-root-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b6640893b24e8d6d93612119d127ba9986706d992fe44c6680c3bae6c440de1","family_id":"qf-principal-square-root-domain","seed":202610031100,"semantic_template":"qf-principal-square-root-domain/{\"domain\":\"reals\",\"operation\":\"sqrt_domain\",\"r\":12}"},"fields":{"title":"Real Domain of the Principal Square Root","stem":"The real principal square root in this exercise is defined only for nonnegative inputs. Does r=12 belong to its domain? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"r=12 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition."},"skills":["Real Domain of the Principal Square Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}