Question 1419: Product Dimension by Componentwise Exponent Addition

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. left=[1, 1, -2], right=[2, 1, 1]. Give (left, right, product 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. left=[1, 1, -2], right=[2, 1, 1]. Give (left, right, product 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

Step-by-step derivation

dimension coordinate order=[L,M,T]

left/base=[1, 1, -2]; right=[2, 1, 1]

formal units L^a M^b T^c

product adds;quotient subtracts;power scales exponents

result exponent L := sum(1, 2)

result exponent L = 3

result exponent M := sum(1, 1)

result exponent M = 2

result exponent T := sum(-2, 1)

result exponent T = -1

[3, 2, -1]

formal dimensional result only

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

([1, 1, -2], [2, 1, 1], [3, 2, -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.