CAT (QA) PYQs

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

Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of $2$. Then, the smallest possible value of $n + m$ is

If $m$ and $n$ are integers such that $(\sqrt{2})^{19} 3^{4} 4^{2} 9^m8^n = 3^n 16^m (\sqrt[4]{64})$ then $m$ is