Question

9．计算：$\sqrt{2+\sqrt{3}}$•$\sqrt{2+\sqrt{2+\sqrt{3}}}$•$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}$•$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$．

Answer 1

分析 利用平方差公式逐步从后面两两结合计算得出答案即可．

解答 解：原式=$\sqrt{2+\sqrt{3}}$•$\sqrt{2+\sqrt{2+\sqrt{3}}}$•$\sqrt{4-2-\sqrt{2+\sqrt{3}}}$
=$\sqrt{2+\sqrt{3}}$•$\sqrt{2+\sqrt{2+\sqrt{3}}}$•$\sqrt{2-\sqrt{2+\sqrt{3}}}$
=$\sqrt{2+\sqrt{3}}$•$\sqrt{4-2-\sqrt{3}}$
=$\sqrt{2+\sqrt{3}}$•$\sqrt{2-\sqrt{3}}$
=4-3
=1．

点评 此题考查二次根式的化简求值，掌握平方差公式是解决问题的关键．