CAT (QA) PYQs

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

If $3^{a}=4,4^{b}=5,5^{c}=6,6^{d}=7,7^{e}=8$ and $8^{f}=9$, then the value of the product $\textit{abcdef }$ 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