Solve for $x$: $2^{2x} = 256^\frac{1}{2}$.
On the left side, I can use the rule that $a^{bc} = (a^b)^c$, and on the right side, I can use the rule that $a^\frac{1}{b} = \sqrt[b]{a}$. So, the equation becomes $(2^2)^x = 2^4$. Now, I can use the rule that $(a^b)^c = a^{bc}$ again, but in reverse, to simplify the left side. The equation becomes $2^{2x} = 2^4$. Since the bases are the same, I can just equate the exponents and solve for x. So, I have $2x = 4$, and dividing both sides by 2 gives me $x = 2$. 