与えられた9個の平方根の計算問題を解きます。

1. 問題の内容

与えられた9個の平方根の計算問題を解きます。

2. 解き方の手順

各問題について、以下の手順で計算します。

(1) $\sqrt{15} \times \sqrt{80}$

$\sqrt{15 \times 80} = \sqrt{15 \times 16 \times 5} = \sqrt{15 \times 5 \times 16} = \sqrt{75 \times 16} = \sqrt{25 \times 3 \times 16} = 5 \times 4 \times \sqrt{3} = 20\sqrt{3}$

(2) $\frac{\sqrt{75}}{2} \times \sqrt{12}$

$\frac{\sqrt{75 \times 12}}{2} = \frac{\sqrt{25 \times 3 \times 4 \times 3}}{2} = \frac{\sqrt{25 \times 4 \times 9}}{2} = \frac{5 \times 2 \times 3}{2} = 15$

(3) $\sqrt{15} \div \frac{\sqrt{3}}{\sqrt{5}}$

$\sqrt{15} \times \frac{\sqrt{5}}{\sqrt{3}} = \sqrt{\frac{15 \times 5}{3}} = \sqrt{5 \times 5} = 5$

(4) $\sqrt{8} \times 2\sqrt{14} \times \sqrt{7}$

$2 \times \sqrt{8 \times 14 \times 7} = 2 \times \sqrt{8 \times 2 \times 7 \times 7} = 2 \times \sqrt{16 \times 49} = 2 \times 4 \times 7 = 56$

(5) $\sqrt{52} \times \sqrt{18} \div \sqrt{39}$

$\frac{\sqrt{52} \times \sqrt{18}}{\sqrt{39}} = \sqrt{\frac{52 \times 18}{39}} = \sqrt{\frac{4 \times 13 \times 9 \times 2}{3 \times 13}} = \sqrt{4 \times 3 \times 2} = \sqrt{4 \times 6} = \sqrt{24} = 2\sqrt{6}$

(6) $\sqrt{27} \div 2\sqrt{45} \times \sqrt{32}$

$\frac{\sqrt{27} \times \sqrt{32}}{2\sqrt{45}} = \frac{\sqrt{9 \times 3 \times 16 \times 2}}{2\sqrt{9 \times 5}} = \frac{3 \times 4 \sqrt{6}}{2 \times 3 \sqrt{5}} = \frac{2\sqrt{6}}{\sqrt{5}} = \frac{2\sqrt{30}}{5}$

(7) $\sqrt{60} \div \sqrt{28} \div \sqrt{15}$

$\frac{\sqrt{60}}{\sqrt{28} \times \sqrt{15}} = \frac{\sqrt{4 \times 15}}{\sqrt{4 \times 7} \times \sqrt{15}} = \frac{2\sqrt{15}}{2\sqrt{7} \times \sqrt{15}} = \frac{1}{\sqrt{7}} = \frac{\sqrt{7}}{7}$

(8) $\sqrt{135} \div \sqrt{5} \times \frac{\sqrt{98}}{\sqrt{3}}$

$\frac{\sqrt{135}}{\sqrt{5}} \times \frac{\sqrt{98}}{\sqrt{3}} = \sqrt{\frac{135}{5}} \times \sqrt{\frac{98}{3}} = \sqrt{27} \times \sqrt{\frac{98}{3}} = \sqrt{9 \times 3 \times \frac{98}{3}} = \sqrt{9 \times 98} = \sqrt{9 \times 49 \times 2} = 3 \times 7 \sqrt{2} = 21\sqrt{2}$

(9) $\frac{2\sqrt{3}}{\sqrt{15}} \div \frac{\sqrt{35}}{5} \times \frac{\sqrt{14}}{4}$

$\frac{2\sqrt{3}}{\sqrt{15}} \times \frac{5}{\sqrt{35}} \times \frac{\sqrt{14}}{4} = \frac{10 \sqrt{3} \sqrt{14}}{4 \sqrt{15} \sqrt{35}} = \frac{5 \sqrt{3} \sqrt{2} \sqrt{7}}{2 \sqrt{3} \sqrt{5} \sqrt{5} \sqrt{7}} = \frac{5 \sqrt{2}}{2 \times 5} = \frac{\sqrt{2}}{2}$

3. 最終的な答え

(1) $20\sqrt{3}$

(2) 15

(3) 5

(4) 56

(5) $2\sqrt{6}$

(6) $\frac{2\sqrt{30}}{5}$

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

(8) $21\sqrt{2}$

(9) $\frac{\sqrt{2}}{2}$