# Question 28277: Principal Square Root of a Perfect Square

[Return to catalogue page 142](https://ocme.evemisslab.com/en/questions/page/142/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r01-power-root-foundations/#q-20261003-r01-power-root-0052)

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.

R0055-28277 · q-20261003-r01-power-root-0052



## Original question and givens

Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.


- G001 — Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.





## 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-principal-square-root-exact, principal_sqrt]

After

derived = [[single_nonnegative_exact_integer_root]]

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 principal ORIGINAL radicand exact midpoint square0 := mul(1, 1)

After

Legacy principal ORIGINAL radicand exact midpoint square0 = 1

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 · impliesBefore

source roles = [Legacy principal ORIGINAL radicand, 1, 0, [0, 2], 1]

After

derived = [1, Yes, 1, 2]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

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





S004 · calculationBefore

Legacy principal ORIGINAL radicand terminal root square := mul(1, 1)

After

Legacy principal ORIGINAL radicand terminal root square = 1

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, S003





S005 · calculationBefore

Legacy principal ORIGINAL radicand terminal successor square := mul(2, 2)

After

Legacy principal ORIGINAL radicand terminal successor square = 4

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, S004





S006 · impliesBefore

source roles = [Legacy principal ORIGINAL radicand, 1, 1, 2]

After

derived = [1, 4]

Conditions: none

Adjacent integer bounds certify floor sqrt=low;an exact root separately needs low²=n,and the principal root is the single nonnegative branch.

Step identifiers and dependenciesRule: Legacy_isqrt_terminal_floor_and_single_nonnegative_branch · Uses: G001, S005





S007 · impliesBefore

source roles = [1]

After

derived = [1, Yes]

Conditions: none

Check root²=r and root>=0.Monotonicity of nonnegative squaring gives uniqueness;two signed equation roots are not the principal-root value.

Step identifiers and dependenciesRule: Legacy_PERFECT_square_exact_NONNEGATIVE_principal_root · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
- 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 principal ORIGINAL radicand terminal successor square with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand terminal root square with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
- S002 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square0 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

1

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: 640e9198a544bbb6396db32066c8eeab28fa43525d369fcc2407b0c291a9d65d
- candidate record SHA-256: 810e6276fd7cf9c4349792583431098e94a27ddde4a0780f35551a1d4a9e9497
- 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-0052.json)