CAT 2024 Slot 1 QA Question 17: The sum of all real values of $k$ for which $\left(\dfrac{1}{8} ight)^k imes \left(\dfrac{1}{32768} ight)^{1/3} = \left(\dfrac{1}{8} ight) imes \left(\dfrac{1}{32768} ight)^{1/k}$ is,

Question

The sum of all real values of $k$ for which $\left(\dfrac{1}{8}ight)^k 	imes \left(\dfrac{1}{32768}ight)^{1/3} = \left(\dfrac{1}{8}ight) 	imes \left(\dfrac{1}{32768}ight)^{1/k}$ is,

Accepted Answer

Write both sides with base $2$.
$\dfrac{1}{8} = 2^{-3}$ and $\dfrac{1}{32768} = 2^{-15}$
Left side = $(2^{-3})^k	imes (2^{-15})^{1/3} = 2^{-3k-5}$
Right side = $2^{-3}	imes (2^{-15})^{1/k} = 2^{-3-15/k}$
Since the bases are equal, equate the exponents.
$-3k - 5 = -3 - \dfrac{15}{k}$
$\Rightarrow 3

The sum of all real values of kkk for which (18)k×(132768)1/3=(18)×(132768)1/k\left(\dfrac{1}{8}\right)^k \times \left(\dfrac{1}{32768}\right)^{1/3} = \left(\dfrac{1}{8}\right) \times \left(\dfrac{1}{32768}\right)^{1/k}(81​)k×(327681​)1/3=(81​)×(327681​)1/k is,

The sum of all real values of $k$ for which $\left(\dfrac{1}{8}\right)^k \times \left(\dfrac{1}{32768}\right)^{1/3} = \left(\dfrac{1}{8}\right) \times \left(\dfrac{1}{32768}\right)^{1/k}$ is,