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

$\sqrt{\frac{q}{p}} = \sqrt{\frac{p}{q}}$

GATE 2024 BM – Question 3

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

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

$q^p = p^q$
$q^p = p^2 q$
$\sqrt{q} = \sqrt{p}$
$\sqrt{\frac{q}{p}} = \sqrt{\frac{p}{q}}$

Practise this question in The GATE Grind →

Show answer and explanation

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

Explanation

Let $t = p/q$. Then $t^t = p^{t-1}$ with $p = tq$, so $t^t = t^{t-1} q^{t-1}$, which gives $t = q^{t-1}$, that is $p = q^{t} = q^{p/q}$. Raising both sides to the power $q$ gives $p^q = q^p$.

More Quantitative Aptitude: Algebra, Geometry and Mensuration questions

GATE 2024 BM Q7 – Three distinct sets of indistinguishable twins are to be seated at a circular table that…
GATE 2024 BM Q10 – During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun…
GATE 2023 BM Q7 – Which function represents the given graph?
GATE 2022 BM Q3 – If r is a root of x²+2x+6=0, find (r+2)(r+3)(r+4)(r+5).
GATE 2021 BM Q4 – Define operators by p⊙q=p−q and p⊕q=p×q. Evaluate (9⊙(6⊕7))⊙(7⊕(6⊙5)).
GATE 2020 BM Q9 – A 4×4 matrix M=[mᵢⱼ] has zero diagonal and mᵢⱼ=−mⱼᵢ. What minimum number of entries…

All Quantitative Aptitude: Algebra, Geometry and Mensuration questions →