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2007 AMC 10A Problems/Problem 20

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Problem

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

\text{(A)}\ 164 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 192 \qquad \text{(D)}\ 194 \qquad \text{(E)}\ 212


Solution

Solution 1

Notice that (a^{k} + a^{-k})^2 = a^{2k} + a^{-2k} + 2. Thus a^4 + a^{-4} = (a^2 + a^{-2})^2 - 2 = [(a + a^{-1})^2 - 2]^2 - 2 = 194\ \mathrm{(D)}.

Solution 2

4a = a^2 + 1. We apply the quadratic formula to get a = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}.

Thus a^4 + a^{-4} = (2+\sqrt{3})^4 + \frac{1}{(2+\sqrt{3})^4} = (2+\sqrt{3})^4 + (2-\sqrt{3})^4 (so it doesn't matter which root of a we use). Using the binomial theorem we can expand this out and collect terms to get 194.

See also

2007 AMC 10A (Problems • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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