Bài 58 (Trang 32 SGK Toán 9, Tập 1):

Rút gọn các biểu thức sau:

$\mathrm{a}) 5\sqrt{\frac{1}{5}} + \frac{1}{2}\sqrt{20} + \sqrt{5};$

$\mathrm{b}) \sqrt{\frac{1}{2}} + \sqrt{4,5} + \sqrt{12,5}.$

$\mathrm{c}) \sqrt{20} - \sqrt{45} + 3\sqrt{8} + \sqrt{72};$

$\mathrm{d}) 0,1.\sqrt{200} +\hspace{0.17em}2.\sqrt{0,08}.\sqrt{50}.$

Hướng dẫn giải:

$\mathrm{a}) 5\sqrt{\frac{1}{5}} + \frac{1}{2}\sqrt{20} + \sqrt{5} = \sqrt{5^2.\frac{1}{5}} + \sqrt{\frac{1}{2^2}.20} + \sqrt{5} = \sqrt{5} + \sqrt{5} + \sqrt{5} = 3\sqrt{5}.$

$$\begin{gathered} \mathrm{b}) \sqrt{\frac{1}{2}} + \sqrt{4,5} + \sqrt{12,5} = \sqrt{\frac{1}{2}} + \sqrt{\frac{9}{2}} + \sqrt{\frac{25}{2}} = \sqrt{\frac{1}{2}} + \sqrt{3^2.\frac{1}{2}} + \sqrt{5^2.\frac{1}{2}} \\ = \sqrt{\frac{1}{2}} + 3\sqrt{\frac{1}{2}} + 5\sqrt{\frac{1}{2}} = 9\sqrt{\frac{1}{2}} = 9\frac{\sqrt{1}}{\sqrt{2}} \\ = 9\frac{\sqrt{2}}{\sqrt{2}.\sqrt{2}} = \frac{9}{2}\sqrt{2}. \end{gathered}$$

$$\begin{gathered} \mathrm{c}) \sqrt{20} - \sqrt{45} + 3\sqrt{18} + \sqrt{72} = \sqrt{4.5} - \sqrt{9.5} + 3\sqrt{9.2} + \sqrt{36.2} \\ = 2\sqrt{5} - 3\sqrt{5} + 9\sqrt{2} + 6\sqrt{2} = -\sqrt{5} + 15\sqrt{5}. \end{gathered}$$

$$\begin{gathered} \mathrm{d}) 0,1.\sqrt{200} + 2.\sqrt{0,08}.\sqrt{50} = 0,1.\sqrt{100.2} + 2\sqrt{\frac{8}{100}} + 0,4.\sqrt{25.2} \\ = \sqrt{2} + 2.\frac{\sqrt{2}}{\sqrt{10}} + 2\sqrt{2} = \sqrt{2} + 0,4\sqrt{2} + 2\sqrt{2} = 3\sqrt{2}. \end{gathered}$$