Can $(5^k+1)(2^k)$ be a perfect power?
I noticed that $(5^k+1)(2^k)=(10^k+2^k)$, and this cannot be a perfect square since the last digit of $(5^k+1)(2^k)$ is either $2$, or $8$ when $k$ is odd, and when $k$ is even, it is equal to $2\pmod{3}$.
$(5^k+1)(2^k)$ is sometimes equal to $0,\pm1\pmod{9}$, so $(5^k+1)(2^k)$ maybe can be a perfect cube.
$(5^k+1)(2^k)$ can be only a perfect 5th power if the last 2 digits of the product is either $24, 64, 68,$ or $76$.
But $(5^k+1)(2^k)$ can be a higher perfect odd prime perfect power?
Using brute force, I tried to checked the values of $k\leq10^{4}$, but no number listed there is a perfect power.
Hint:$k \in \mathbb{Z}^{+}$ and $(5^k+1)(2^k)=(10^k)+(2^k)$, so $2^{k+1}\mid(5^k+1)(2^k)$.