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Ted's Mistake

Source: 2022 AMC 10A #11

November 11, 2022
AMCAMC 102022 AMC2022 AMC 10aalgebra

Problem Statement

Ted mistakenly wrote 2m⋅140962^m \cdot \sqrt{\frac{1}{4096}} as 2⋅14096m2\cdot \sqrt[m]{\frac{1}{4096}}. What is the sum of all real numbers mm for which these two expressions have the same value?
<spanclass=′latex−bold′>(A)</span>5<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>7<spanclass=′latex−bold′>(D)</span>8<spanclass=′latex−bold′>(E)</span>9<span class='latex-bold'>(A) </span>5\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>7\qquad<span class='latex-bold'>(D) </span>8\qquad<span class='latex-bold'>(E) </span>9
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