1979 AMC 12 Problem 19

Attempt Problem 19 of the 1979 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1979 AMC 12 solutions, or check the answer key.

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19.

Find the sum of the squares of all real numbers satisfying the equation $x^{256}-256^{32}=0.$

$8$

$128$

$512$

$65{,}536$

$2(256^{32})$

Answer: A

Concepts:exponent perfect power zero product property

Difficulty rating: 1650

Small Hint:

Rewrite $256^{32}$ as a single power with exponent $256$

Big Hint:

Determine the two real values whose $256$ th powers equal $2^{256}$

Solution:

Since $256=2^8,$ $256^{32}=2^{256}.$ Thus $(\frac{x}{2})^{256}=1.$ The real solutions are $x=2$ and $x=-2,$ and the sum of their squares is $4+4=8.$

Therefore, the correct answer is A.

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