与えられた6つの式を計算し、分母を有理化して簡略化します。 (1) $\frac{1}{\sqrt{5}} + \sqrt{5}$ (2) $\sqrt{\frac{1}{20}} + \frac{\sqrt{5}}{5}$ (3) $\frac{6 + \sqrt{3}}{\sqrt{3}}$ (4) $\frac{3\sqrt{2}}{2\sqrt{3}} - 2\sqrt{6}$ (5) $3\sqrt{5} - \sqrt{\frac{1}{5}}$ (6) $(\sqrt{8}+2)(\sqrt{8}-1) + \frac{5\sqrt{6}}{\sqrt{3}} \times 2$

はい、承知いたしました。与えられた問題を解いていきます。

1. 問題の内容

与えられた6つの式を計算し、分母を有理化して簡略化します。

(1) $\frac{1}{\sqrt{5}} + \sqrt{5}$

(2) $\sqrt{\frac{1}{20}} + \frac{\sqrt{5}}{5}$

(3) $\frac{6 + \sqrt{3}}{\sqrt{3}}$

(4) $\frac{3\sqrt{2}}{2\sqrt{3}} - 2\sqrt{6}$

(5) $3\sqrt{5} - \sqrt{\frac{1}{5}}$

(6) $(\sqrt{8}+2)(\sqrt{8}-1) + \frac{5\sqrt{6}}{\sqrt{3}} \times 2$

2. 解き方の手順

(1)

$\frac{1}{\sqrt{5}} + \sqrt{5} = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} + \sqrt{5} = \frac{\sqrt{5}}{5} + \sqrt{5} = \frac{\sqrt{5} + 5\sqrt{5}}{5} = \frac{6\sqrt{5}}{5}$

(2)

$\sqrt{\frac{1}{20}} + \frac{\sqrt{5}}{5} = \frac{1}{\sqrt{20}} + \frac{\sqrt{5}}{5} = \frac{1}{2\sqrt{5}} + \frac{\sqrt{5}}{5} = \frac{1}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} + \frac{\sqrt{5}}{5} = \frac{\sqrt{5}}{10} + \frac{\sqrt{5}}{5} = \frac{\sqrt{5} + 2\sqrt{5}}{10} = \frac{3\sqrt{5}}{10}$

(3)

$\frac{6 + \sqrt{3}}{\sqrt{3}} = \frac{6 + \sqrt{3}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3} + 3}{3} = \frac{3(2\sqrt{3} + 1)}{3} = 2\sqrt{3} + 1$

(4)

$\frac{3\sqrt{2}}{2\sqrt{3}} - 2\sqrt{6} = \frac{3\sqrt{2}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} - 2\sqrt{6} = \frac{3\sqrt{6}}{6} - 2\sqrt{6} = \frac{\sqrt{6}}{2} - 2\sqrt{6} = \frac{\sqrt{6} - 4\sqrt{6}}{2} = \frac{-3\sqrt{6}}{2}$

(5)

$3\sqrt{5} - \sqrt{\frac{1}{5}} = 3\sqrt{5} - \frac{1}{\sqrt{5}} = 3\sqrt{5} - \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = 3\sqrt{5} - \frac{\sqrt{5}}{5} = \frac{15\sqrt{5} - \sqrt{5}}{5} = \frac{14\sqrt{5}}{5}$

(6)

$(\sqrt{8}+2)(\sqrt{8}-1) + \frac{5\sqrt{6}}{\sqrt{3}} \times 2 = (2\sqrt{2}+2)(2\sqrt{2}-1) + \frac{5\sqrt{2}\sqrt{3}}{\sqrt{3}} \times 2 = (8 - 2\sqrt{2} + 4\sqrt{2} - 2) + 10\sqrt{2} = 6 + 2\sqrt{2} + 10\sqrt{2} = 6 + 12\sqrt{2}$

3. 最終的な答え

(1) $\frac{6\sqrt{5}}{5}$

(2) $\frac{3\sqrt{5}}{10}$

(3) $2\sqrt{3} + 1$

(4) $-\frac{3\sqrt{6}}{2}$

(5) $\frac{14\sqrt{5}}{5}$

(6) $6 + 12\sqrt{2}$