2022 AMC 10A Problem 11

Question

Ted mistakenly wrote

2^m\cdot\sqrt{\dfrac{1}{4096}}

as

2\cdot\sqrt[m]{\dfrac{1}{4096}}.

What is the sum of all real numbers

m

for which these two expressions have the same value?

5

6

7

8

9

Rewrite

4096

as a power of

2

After matching powers of

2,

solve the resulting quadratic equation in m

We can rewrite

4096

as

2^{12},

so

\dfrac{1}{4096} = 2^{-12}.

Then if we equate the given expressions, we get

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

Equating the exponents, we get

m - 6 = 1 + \dfrac{-12}{m}.

Multiplying by

m,

we get

m^2 - 6m = m - 12

and so

m^2 - 7m + 12 = 0

(m-4)(m-3)

m=4,~m=3

Therefore, we can see that the sum of the solutions is

7.

Thus, C is the correct answer.

Problem 11 in other years