2007 AMC 10A Problem 20

Attempt Problem 20 of the 2007 AMC 10A 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 2007 AMC 10A solutions, or check the answer key.

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20.

Suppose that the number

a

satisfies the equation

4 = a + a^{-1}.

What is the value of

a^4 + a^{-4}?

164

172

192

194

212

Squaring

a + a^{-1}

gives

a^2 + 2 + a^{-2}

Square

a^2 + a^{-2}

and subtract the middle term

2

Squaring

a + a^{-1} = 4

gives

a^2 + 2 + a^{-2} = 16,

a^2 + a^{-2} = 14.

Squaring again gives

a^4 + 2 + a^{-4} = 196,

a^4 + a^{-4} = 194.

Thus, the correct answer is D.

Problem 20 in other years