Exponent Simplification

Algebra Level 3

$\Large{\text{Is }\left(2^\color{#D61F06}{a}\right)^{\color{#3D99F6}{b}} \text{ } = \text{ } 2^{\color{#D61F06}{a}+\color{#3D99F6}{b}} \quad \text{ or } \quad 2^{\color{#D61F06}{a}\color{#3D99F6}{b}} \text{ ?}}$

To see which simplification is correct, you plug in specific numbers $\color{#D61F06}{a}=A$ and $\color{#3D99F6}{b}=B$ , but somehow both proposed simplifications give the same answer: $64.$ Find $\left(A-B\right)^2.$