2007 AMC 10A Problem 20 — Solution | LIVE by Po-Shen Loh

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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 $a^4 + a^{-4}?$

$164$

$172$

$192$

$194$

$212$

Answer: D

Concepts:algebraic manipulation symmetry (algebra)

Difficulty rating: 1460

Solution:

Squaring $a + a^{-1} = 4$ gives $a^2 + 2 + a^{-2} = 16,$ so $a^2 + a^{-2} = 14.$

Squaring again gives $a^4 + 2 + a^{-4} = 196,$ so $a^4 + a^{-4} = 194.$

Thus, the correct answer is D.

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