Question

7．四个数$\sqrt{2-\sqrt{3}}$，$\sqrt{2-\sqrt{2-\sqrt{3}}}$，$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}}$，$\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}$的乘积为（　　）

• A． 2+$\sqrt{3}$
• B． 2
• C． 1
• D． 2-$\sqrt{3}$

Answer 1

分析 根据题意列出算式，利用二次根式乘法法则及平方差公式化简，计算即可得到结果．

解答 解：$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}}$•$\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}$•$\sqrt{2-\sqrt{2-\sqrt{3}}}$•$\sqrt{2-\sqrt{3}}$
=$\sqrt{（2-\sqrt{2-\sqrt{2-\sqrt{3}}}）（2+\sqrt{2-\sqrt{2-\sqrt{3}}}）}$•$\sqrt{2-\sqrt{2-\sqrt{3}}}$•$\sqrt{2-\sqrt{3}}$
=$\sqrt{2+\sqrt{2-\sqrt{3}}}$•$\sqrt{2-\sqrt{2-\sqrt{3}}}$•$\sqrt{2-\sqrt{3}}$
=$\sqrt{2+\sqrt{3}}$•$\sqrt{2-\sqrt{3}}$
=$\sqrt{4-3}$
=1．
故选C

点评 此题考查了实数的运算，以及平方差公式，熟练掌握公式及法则是解本题的关键．