AMC 10 2022a Problem 11

Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value? $\textbf{(A) } 5 \textbf{(B) } 6 \textbf{(C) } 7 \textbf{(D) } 8 \textbf{(E) } 9$