1990 AMC 12 Problem 22

Attempt Problem 22 of the 1990 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 1990 AMC 12 solutions, or check the answer key.

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22.

If the six solutions of

x^6=-64

are written in the form

a+bi,

where

a

and

b

are real, then the product of those solutions with

a\gt0

-2

0

2i

4

16

Write

-64

in polar form and list its six sixth roots

The roots with positive real part form a conjugate pair

The roots have modulus

2

and arguments

\frac\pi6+\frac{k\pi}{3},\qquad k=0,1,\ldots,5.

The roots with positive real part have arguments

\frac{\pi}{6}

and

\frac{11\pi}{6}.

They are conjugates of modulus

2,

so their product is

2^2=4.

Thus the correct answer is D.

Problem 22 in other years