CAT 2022 Slot 3 QA Question 10: If $\left(\sqrt{\dfrac{7}{5}}\right)^{3x-y} = \dfrac{875}{2401}$ and $\left(\dfrac{4a}{b}\right)^{6x-y} = \left(\dfrac{2a}{b}\right)^{y-6x}$, for all non-zero real values of a and b, then the value of

Question

If $\left(\sqrt{\dfrac{7}{5}}\right)^{3x-y} = \dfrac{875}{2401}$ and $\left(\dfrac{4a}{b}\right)^{6x-y} = \left(\dfrac{2a}{b}\right)^{y-6x}$, for all non-zero real values of a and b, then the value of

Accepted Answer

First simplify the right side of the first equation.
$\dfrac{875}{2401} = \dfrac{5^3}{7^3} = \left(\dfrac{7}{5}\right)^{-3}$
Also,
$\left(\sqrt{\dfrac{7}{5}}\right)^{3x - y} = \left(\dfrac{7}{5}\right)^{\dfrac{3x - y}{2}}$
Hence,
$\dfrac{3x - y}{2} = -3$
$3x - y = -6$
For the second equation, let $m