1. We are given the exponential function $y = a^x$ and a point on the graph $(2, 64)$. 2. Substitute $x = 2$ and $y = 64$ into the equation: $$64 = a^2$$ 3. To find $a$, take the square root of both sides: $$a = \sqrt{64}$$ 4. Since $a > 0$, we take the positive root: $$a = 8$$ **Final answer:** $a = 8$