2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 11 Easy

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?

  • A.

    5

  • B.

    6

  • C.

    7

  • D.

    8

  • E.

    9

Answer:C

2^m \cdot \left(\frac {1} {4096} \right)^\frac 1 2=2 \cdot \left(\frac {1} {4096} \right)^\frac 1 m

2^m \cdot \left(2^{-12} \right)^\frac1 2=2 \cdot \left(2^{-12} \right)^\frac1 m

2^m \cdot 2^{-6}=2 \cdot 2^{-\frac{12}{m}}

m-6=1-\frac{12}{m}

m+\frac{12}{m}=7

m^2-7m+12=0

Using Vieta's formula, sum of roots =7.

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