与えられた分数の分母を有理化する問題です。以下の7つの問題があります。 (1) $\frac{7}{\sqrt{14}}$ (2) $\frac{\sqrt{12}}{\sqrt{18}}$ (3) $\frac{30}{\sqrt{60}}$ (4) $\frac{\sqrt{20}}{\sqrt{32}}$ (5) $\frac{5\sqrt{2}}{2\sqrt{6}}$ (6) $\frac{\sqrt{15}}{5\sqrt{10}}$ (7) $\frac{3}{2}\sqrt{\frac{2}{3}}$

1. 問題の内容

与えられた分数の分母を有理化する問題です。以下の7つの問題があります。

(1) $\frac{7}{\sqrt{14}}$

(2) $\frac{\sqrt{12}}{\sqrt{18}}$

(3) $\frac{30}{\sqrt{60}}$

(4) $\frac{\sqrt{20}}{\sqrt{32}}$

(5) $\frac{5\sqrt{2}}{2\sqrt{6}}$

(6) $\frac{\sqrt{15}}{5\sqrt{10}}$

(7) $\frac{3}{2}\sqrt{\frac{2}{3}}$

2. 解き方の手順

分母を有理化するには、分母と分子に適切な数を掛けて、分母から根号を取り除きます。

(1) $\frac{7}{\sqrt{14}} = \frac{7\sqrt{14}}{\sqrt{14} \cdot \sqrt{14}} = \frac{7\sqrt{14}}{14} = \frac{\sqrt{14}}{2}$

(2) $\frac{\sqrt{12}}{\sqrt{18}} = \sqrt{\frac{12}{18}} = \sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{6}}{3}$

(3) $\frac{30}{\sqrt{60}} = \frac{30}{\sqrt{4 \cdot 15}} = \frac{30}{2\sqrt{15}} = \frac{15}{\sqrt{15}} = \frac{15\sqrt{15}}{\sqrt{15} \cdot \sqrt{15}} = \frac{15\sqrt{15}}{15} = \sqrt{15}$

(4) $\frac{\sqrt{20}}{\sqrt{32}} = \sqrt{\frac{20}{32}} = \sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2\sqrt{2}} = \frac{\sqrt{5} \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{10}}{4}$

(5) $\frac{5\sqrt{2}}{2\sqrt{6}} = \frac{5\sqrt{2}}{2\sqrt{2 \cdot 3}} = \frac{5\sqrt{2}}{2\sqrt{2}\sqrt{3}} = \frac{5}{2\sqrt{3}} = \frac{5\sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{5\sqrt{3}}{6}$

(6) $\frac{\sqrt{15}}{5\sqrt{10}} = \frac{\sqrt{3 \cdot 5}}{5\sqrt{2 \cdot 5}} = \frac{\sqrt{3}\sqrt{5}}{5\sqrt{2}\sqrt{5}} = \frac{\sqrt{3}}{5\sqrt{2}} = \frac{\sqrt{3} \cdot \sqrt{2}}{5\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{6}}{10}$

(7) $\frac{3}{2}\sqrt{\frac{2}{3}} = \frac{3}{2} \cdot \frac{\sqrt{2}}{\sqrt{3}} = \frac{3}{2} \cdot \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{6}}{2 \cdot 3} = \frac{\sqrt{6}}{2}$

3. 最終的な答え

(1) $\frac{\sqrt{14}}{2}$

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

(3) $\sqrt{15}$

(4) $\frac{\sqrt{10}}{4}$

(5) $\frac{5\sqrt{3}}{6}$

(6) $\frac{\sqrt{6}}{10}$

(7) $\frac{\sqrt{6}}{2}$