For positive integers p and q, with pq 1, (pq)p/q = p(p/q - 1). Then,

GATE 2024 CH – Question 3

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

For positive integers $p$ and $q$, with $\frac{p}{q} \ne 1$, $\left(\frac{p}{q}\right)^{p/q} = p^{(p/q - 1)}$. Then,

$q^p = p^q$
$q^p = p^{2q}$
$\sqrt{q} = \sqrt{p}$
$\sqrt[q]{p} = \sqrt[p]{q}$

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Correct answer: (A) $q^p = p^q$

Explanation

Take logarithms: $\frac{p}{q}(\ln p - \ln q) = \left(\frac{p}{q} - 1\right)\ln p$. So $\frac{p}{q}\ln p - \frac{p}{q}\ln q = \frac{p}{q}\ln p - \ln p$, which gives $\frac{p}{q}\ln q = \ln p$. Multiplying by $q$: $p\ln q = q\ln p$, that is $q^p = p^q$.