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Question 28237: Sign of a Natural-Number Power

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0055-28237 · q-20261003-r01-power-root-0037

Original question and givens

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?

  • G001 — 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?

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-natural-power-sign, power_sign]

After

derived = [[power_sign_label]]

Conditions: none

List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.

Step identifiers and dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Legacy power ORIGINAL multiply1 := mul(1, 1)

After

Legacy power ORIGINAL multiply1 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Legacy power ORIGINAL multiply2 := mul(1, 1)

After

Legacy power ORIGINAL multiply2 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Legacy power ORIGINAL multiply3 := mul(1, 1)

After

Legacy power ORIGINAL multiply3 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Legacy power ORIGINAL multiply4 := mul(1, 1)

After

Legacy power ORIGINAL multiply4 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, 4, empty_product]

After

derived = [1, positive]

Conditions: none

The n0 empty product is1,including0^0=1 in THIS algebraic setting,not a limit rule.For n>0 a zero base gives0;otherwise sign follows base sign and exponent parity.All BigInt results remain exact without derived SAFE/1e8 clipping.

Step identifiers and dependencies

Rule: Legacy_power_ALL_n_products_EMPTYproduct_n0_and_sign_parity · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S005 · premise_and_result_audit — Recompute Legacy power ORIGINAL multiply4 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Legacy power ORIGINAL multiply3 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Legacy power ORIGINAL multiply2 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Legacy power ORIGINAL multiply1 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

positive

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 171ba754dbf39a46e77a7e406ab81db143728421954ee85df9dca95a9cb75e1a
  • candidate record SHA-256: 7fcc140510ebe43962cfd2f6fa8fa7e85e4bb64c3e853c7ea19080d9544a0a1d
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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