Problem 92

For a positive integer to be a perfect square, all the primes in its prime factorization must have an even exponent. With a quick glance at the answer choices, we can eliminate options
$\textbf{(A)}$ because $5^5$ is an odd power
$\textbf{(B)}$ because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power
$\textbf{(D)}$ because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power, and
$\textbf{(E)}$ because $5^5$ is an odd power.
This leaves option $\textbf{(C)},$ in which $4^5=(2^{2})^{5}=2^{10}$, and since $10, 4,$ and $6$ are all even, $\textbf{(C)}$ is a perfect square. Thus, our answer is $\boxed{\textbf{(C) } 4^4 5^4 6^6}$.