For positive integers p and q, with p/q != 1, if (p/q)^(p/q) = p^(1/q), then which of the following must be TRUE? A. p^(p-1) = q^p B. p^q = q^p C. sqrt(q) = sqrt(p) D. sqrt(q) = p

GATE 2024 · General Aptitude · Fractions · medium

Answer: The relationship p^(p-1) = q^p must be TRUE, which is option A.

1. Raise both sides of the equation to the power q: Left: (p/q)^((p/q)*q) = (p/q)^p = p^p/q^p. Right: p^((1/q)*q) = p^1 = p.
2. Set the two sides equal and rearrange: Multiply both sides by q^p: p^p = p * q^p. Divide both sides by p: p^(p-1) = q^p.
3. Match to the answer options: This matches option A directly. Options B, C, D do not follow from this identity in general.