AMC10 2011 A

AMC10 2011 A · Q16

AMC10 2011 A · Q16. It mainly tests Exponents & radicals.

Which of the following is equal to $\sqrt{9 -6\sqrt{2}} + \sqrt{9 + 6\sqrt{2}}$?

以下哪一项等于 $\sqrt{9 -6\sqrt{2}} + \sqrt{9 + 6\sqrt{2}}$？

(A) $3\sqrt{2}$ $3\sqrt{2}$

(B) 2$\sqrt{6}$ 2$\sqrt{6}$

(D) 3$\sqrt{3}$ 3$\sqrt{3}$

(E) 6 6

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let $k=\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$. Squaring both sides and simplifying results in $k^2=9-6\sqrt{2}+2\sqrt{(9-6\sqrt{2})(9+6\sqrt{2})}+9+6\sqrt{2}$ $=18+2\sqrt{81-72}$ $=18+2\sqrt{9}$ $=24$ Because $k>0$, $k=2\sqrt{6}$.

答案（B）：设 $k=\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$。两边平方并化简得到 $k^2=9-6\sqrt{2}+2\sqrt{(9-6\sqrt{2})(9+6\sqrt{2})}+9+6\sqrt{2}$ $=18+2\sqrt{81-72}$ $=18+2\sqrt{9}$ $=24$ 因为 $k>0$，所以 $k=2\sqrt{6}$。

Topics

AL.008 Exponents & radicals