# Question 28203: Exact Natural-Number Power

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0055-28203 · q-20261003-r01-power-root-0003



## 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. Compute (-10)^2 exactly.


- G001 — 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.





## 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.


- [mko-power-root](https://ocme.evemisslab.com/en/mko/mko-power-root/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-natural-power-exact, natural_power]

After

derived = [[exact_integer_power]]

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 dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001





S002 · calculationBefore

Legacy power ORIGINAL multiply1 := mul(1, -10)

After

Legacy power ORIGINAL multiply1 = -10

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

Legacy power ORIGINAL multiply2 := mul(-10, -10)

After

Legacy power ORIGINAL multiply2 = 100

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [-10, 2, empty_product]

After

derived = [100, 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 dependenciesRule: Legacy_power_ALL_n_products_EMPTYproduct_n0_and_sign_parity · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
- 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

100

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: 8ab5d0ba4bfb704e2f255b197253d4e7b60d41b32f4acccca8183080727faaa3
- candidate record SHA-256: 9c6e0138f35ecdfad9d0a2b63f8283b5b66bc7a52e11b42befa9dbac2f4054e8
- semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
- question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r01-power-root-0003.json)