AMC10 2007 A
AMC10 2007 A · Q20
AMC10 2007 A · Q20. It mainly tests Exponents & radicals.
Suppose that the number $a$ satisfies the equation $4 = a + a^{-1}$. What is the value of $a^4 + a^{-4}$?
假设数 $a$ 满足方程 $4 = a + a^{-1}$。$a^4 + a^{-4}$ 的值是多少？
(A)
164
164
(B)
172
172
(C)
192
192
(D)
194
194
(E)
212
212
Answer
Correct choice: (D)
正确答案：（D）
Solution
Answer (D): Squaring each side of the equation $4=a+a^{-1}$ gives
$16=a^{2}+2a\cdot a^{-1}+(a^{-1})^{2}=a^{2}+2+a^{-2}$, so $14=a^{2}+a^{-2}$.
Squaring again gives
$196=a^{4}+2a^{2}\cdot a^{-2}+(a^{-2})^{2}=a^{4}+2+a^{-4}$, so $194=a^{4}+a^{-4}$.
答案（D）：将方程 $4=a+a^{-1}$ 的两边平方得
$16=a^{2}+2a\cdot a^{-1}+(a^{-1})^{2}=a^{2}+2+a^{-2}$，所以 $14=a^{2}+a^{-2}$。
再平方一次得
$196=a^{4}+2a^{2}\cdot a^{-2}+(a^{-2})^{2}=a^{4}+2+a^{-4}$，所以 $194=a^{4}+a^{-4}$。
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