Nice Base Problem

$\begin{aligned} \sqrt[a]{10000_a} & = 10_a \\ \sqrt[a]{a^4_{10}} & = a_{10} \\ \implies a & = \boxed 4 \end{aligned}$

We can write the right-hand side expression as $a^{\frac{4}{a}}$ , and set it equal to $a$ . Raising both sides to the power of $a$ gives us $a^4 = a^a$ , and therefore $a = 4$ .