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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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})

Rewrite

256^{32}

as a single power with exponent

256

Determine the two real values whose

256

th powers equal

2^{256}

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.

Problem 19 in other years