【习题解析】 如何解决双重根式化简

习题完整解析#

1. 判断 $\sqrt{5+2\sqrt{5}}$ 能否被化简：若能，则写出化简后的结果；若不能，请说明理由.

   $解：由题，知\Delta=5^2-2^2\times5=5$

   $\because5不是可开尽方的有理数$

   $\therefore原式不能被化简$

2. 判断 $\sqrt{37-12\sqrt{7}}$ 能否被化简：若能，则写出化简后的结果；若不能，请说明理由.

   $解：由题，知\Delta=37^2-12^2\times7=361=19^2$

   $\begin{aligned}\therefore 原式&=\sqrt{\frac{37+19}{2}}-\sqrt{\frac{37-19}{2}}\\&=\sqrt{\frac{56}{2}}-\sqrt{\frac{18}{2}}\\&=2\sqrt{7}-3\end{aligned}$

3. 已知 $\cos\theta=\frac{3\sqrt{2}}{5}$，运用公式 $\tan\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$，计算 $\tan(90\degree -\frac{\theta}{2})$ 的值.

   $\begin{aligned}解：由题，得\tan{\frac{\theta}{2}}&=\sqrt{\frac{1-\frac{3\sqrt{2}}{5}}{1+\frac{3\sqrt{2}}{5}}}\\&=\sqrt{\frac{5-3\sqrt{2}}{5+3\sqrt{2}}}\\&=\sqrt{\frac{43-30\sqrt{2}}{7}}\\&=\sqrt{\frac{1}{7}}\times\sqrt{43-30\sqrt{2}}\end{aligned}$

   $\therefore \Delta=43^2-900\times2=49=7^2$

   $\begin{aligned}\therefore \tan{\frac{\theta}{2}}&=\sqrt{\frac{1}{7}}\times(\sqrt{\frac{43+7}{2}}-\sqrt{\frac{43-7}{2}})\\&=\sqrt{\frac{1}{7}}(5-3\sqrt{2})\\&=\frac{5-3\sqrt{2}}{\sqrt{7}}\end{aligned}$

   $\begin{aligned}\therefore \tan{(90\degree-\frac{\theta}{2})}&=\frac{1}{\tan{\frac{\theta}{2}}}\\&=\frac{\sqrt{7}}{5-3\sqrt{2}}\\&=\frac{5\sqrt{7}+3\sqrt{14}}{7}\end{aligned}$