# Question 1450: Quotient Dimension with Ordered Exponent Subtraction [Return to catalogue page 8](https://ocme.evemisslab.com/en/questions/page/8/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r05-dimensional-analysis-foundations/#q-20261013-advance-20261020-r05-dimensional-analysis-0050) 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. R0001-01450 · q-20261013-advance-20261020-r05-dimensional-analysis-0050 ## Original question and givens Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 2, 3], denominator=[-1, 2, 2]. Give (numerator vector, denominator vector, quotient exponent vector). - G001 — Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 2, 3], denominator=[-1, 2, 2]. Give (numerator vector, denominator vector, quotient exponent vector). ## 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-dimensional-analysis](https://ocme.evemisslab.com/en/mko/mko-dimensional-analysis/) ## Step-by-step derivation S001 · impliesBefore dimension coordinate order=[L,M,T] After left/base=[0, 2, 3]; right=[-1, 2, 2] Conditions: none These are dimension exponents,not measured magnitudes;the order is length,mass,time. Step identifiers and dependenciesRule: dimension_order · Uses: G001 S002 · impliesBefore formal units L^a M^b T^c After product adds;quotient subtracts;power scales exponents Conditions: none Apply exponent laws componentwise in the fixed base dimensions. Step identifiers and dependenciesRule: dimension_exponent_laws · Uses: G001, S001 S003 · calculationBefore result exponent L := sub(0, -1) After result exponent L = 1 Conditions: none Operate at the same base-dimension position,not across different coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore result exponent M := sub(2, 2) After result exponent M = 0 Conditions: none Operate at the same base-dimension position,not across different coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore result exponent T := sub(3, 2) After result exponent T = 1 Conditions: none Operate at the same base-dimension position,not across different coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore [1, 0, 1] After formal dimensional result only Conditions: none Negative exponents denote reciprocal dimensions;an actual quotient or negative power still needs defined magnitude arithmetic,and dimensions alone do not prove a nonzero denominator. Step identifiers and dependenciesRule: dimension_not_magnitude · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute result exponent T with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute result exponent M with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute result exponent L with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 2, 3], [-1, 2, 2], [1, 0, 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: fda939ebc85d003dec5f80f148664b3506c871afacf6b6faf8b45c7205133522 - candidate record SHA-256: c8ed1bb553cf1f099f0af242b8fdbeb840cb1b08d1fb9b37fafe1f82471a6485 - semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc - question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r05-dimensional-analysis-0050.json)