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Question 28295: Principal Square Root of a Perfect Square

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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R0055-28295 · q-20261003-r01-power-root-0070

Original question and givens

Using the principal square root on the nonnegative reals, find the exact nonnegative value of √361.

  • G001 — Using the principal square root on the nonnegative reals, find the exact nonnegative value of √361.

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-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 dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square0 := mul(181, 181)

After

Legacy principal ORIGINAL radicand exact midpoint square0 = 32761

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 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 0, [0, 362], 181]

After

derived = [32761, No, 0, 181]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S002

S004 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square1 := mul(90, 90)

After

Legacy principal ORIGINAL radicand exact midpoint square1 = 8100

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 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 1, [0, 181], 90]

After

derived = [8100, No, 0, 90]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S004

S006 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square2 := mul(45, 45)

After

Legacy principal ORIGINAL radicand exact midpoint square2 = 2025

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, S005

S007 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 2, [0, 90], 45]

After

derived = [2025, No, 0, 45]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S006

S008 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square3 := mul(22, 22)

After

Legacy principal ORIGINAL radicand exact midpoint square3 = 484

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, S007

S009 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 3, [0, 45], 22]

After

derived = [484, No, 0, 22]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S008

S010 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square4 := mul(11, 11)

After

Legacy principal ORIGINAL radicand exact midpoint square4 = 121

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, S009

S011 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 4, [0, 22], 11]

After

derived = [121, Yes, 11, 22]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S010

S012 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square5 := mul(16, 16)

After

Legacy principal ORIGINAL radicand exact midpoint square5 = 256

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, S011

S013 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 5, [11, 22], 16]

After

derived = [256, Yes, 16, 22]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S012

S014 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square6 := mul(19, 19)

After

Legacy principal ORIGINAL radicand exact midpoint square6 = 361

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, S013

S015 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 6, [16, 22], 19]

After

derived = [361, Yes, 19, 22]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S014

S016 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square7 := mul(20, 20)

After

Legacy principal ORIGINAL radicand exact midpoint square7 = 400

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, S015

S017 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 7, [19, 22], 20]

After

derived = [400, No, 19, 20]

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 dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S016

S018 · calculation

Before

Legacy principal ORIGINAL radicand terminal root square := mul(19, 19)

After

Legacy principal ORIGINAL radicand terminal root square = 361

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, S017

S019 · calculation

Before

Legacy principal ORIGINAL radicand terminal successor square := mul(20, 20)

After

Legacy principal ORIGINAL radicand terminal successor square = 400

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, S018

S020 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 361, 19, 20]

After

derived = [361, 400]

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 dependencies

Rule: Legacy_isqrt_terminal_floor_and_single_nonnegative_branch · Uses: G001, S019

S021 · implies

Before

source roles = [361]

After

derived = [19, 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 dependencies

Rule: Legacy_PERFECT_square_exact_NONNEGATIVE_principal_root · Uses: G001, S020

Reverse checks

  • S021 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S020 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S019 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand terminal successor square with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand terminal root square with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S016 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square7 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S014 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square6 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S012 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square5 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S010 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square4 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S008 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square3 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute Legacy principal ORIGINAL radicand exact midpoint square2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S004 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square1 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

19

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: aa15d6714e9f1259ef9b3035d9519538c8eba01cf384f131c5f183432e91509f
  • candidate record SHA-256: 6fccb4f51649ffaec6c9a568c29bee32262b4778ad645e03aca40aa17be4f5ad
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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