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2007 AMC 10A Problem 20

Problem 20 of 25HarderAlgebra

Suppose that the number aa satisfies the equation 4=a+a−1.4 = a + a^{-1}. What is the value of a4+a−4?a^4 + a^{-4}?

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Solution

Squaring a+a−1=4a + a^{-1} = 4 gives a2+2+a−2=16,a^2 + 2 + a^{-2} = 16, so a2+a−2=14.a^2 + a^{-2} = 14. Squaring again gives a4+2+a−4=196,a^4 + 2 + a^{-4} = 196, so a4+a−4=194.a^4 + a^{-4} = 194. Thus, the correct answer is D.

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Concepts: algebraic manipulation · symmetry (algebra)

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.