与えられた8つの計算問題を解きます。これらの問題は、平方根の掛け算と割り算を含みます。

1. 問題の内容

与えられた8つの計算問題を解きます。これらの問題は、平方根の掛け算と割り算を含みます。

2. 解き方の手順

各問題ごとに手順を以下に示します。平方根の中を簡単にできる場合は簡単にします。

(1) $\sqrt{32} \times \sqrt{20}$

$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

$4\sqrt{2} \times 2\sqrt{5} = 8\sqrt{10}$

(2) $\sqrt{6} \times \sqrt{42}$

$\sqrt{6} \times \sqrt{42} = \sqrt{6 \times 42} = \sqrt{6 \times 6 \times 7} = 6\sqrt{7}$

(3) $2\sqrt{15} \times 3\sqrt{3}$

$2\sqrt{15} \times 3\sqrt{3} = 6\sqrt{15 \times 3} = 6\sqrt{45} = 6\sqrt{9 \times 5} = 6 \times 3\sqrt{5} = 18\sqrt{5}$

(4) $(-4\sqrt{2}) \times 2\sqrt{10}$

$(-4\sqrt{2}) \times 2\sqrt{10} = -8\sqrt{2 \times 10} = -8\sqrt{20} = -8\sqrt{4 \times 5} = -8 \times 2\sqrt{5} = -16\sqrt{5}$

(5) $\sqrt{3} \div \sqrt{10}$

$\sqrt{3} \div \sqrt{10} = \frac{\sqrt{3}}{\sqrt{10}} = \frac{\sqrt{3}}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{30}}{10}$

(6) $3\sqrt{5} \div \sqrt{32}$

$3\sqrt{5} \div \sqrt{32} = \frac{3\sqrt{5}}{\sqrt{32}} = \frac{3\sqrt{5}}{\sqrt{16 \times 2}} = \frac{3\sqrt{5}}{4\sqrt{2}} = \frac{3\sqrt{5}}{4\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{10}}{4 \times 2} = \frac{3\sqrt{10}}{8}$

(7) $\sqrt{50} \div \sqrt{6}$

$\sqrt{50} \div \sqrt{6} = \frac{\sqrt{50}}{\sqrt{6}} = \frac{\sqrt{25 \times 2}}{\sqrt{6}} = \frac{5\sqrt{2}}{\sqrt{6}} = \frac{5\sqrt{2}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{5\sqrt{12}}{6} = \frac{5\sqrt{4 \times 3}}{6} = \frac{5 \times 2\sqrt{3}}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3}$

(8) $\sqrt{32} \div (-\sqrt{10})$

$\sqrt{32} \div (-\sqrt{10}) = \frac{\sqrt{32}}{-\sqrt{10}} = -\frac{\sqrt{32}}{\sqrt{10}} = -\frac{\sqrt{16 \times 2}}{\sqrt{10}} = -\frac{4\sqrt{2}}{\sqrt{10}} = -\frac{4\sqrt{2}}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = -\frac{4\sqrt{20}}{10} = -\frac{4\sqrt{4 \times 5}}{10} = -\frac{4 \times 2\sqrt{5}}{10} = -\frac{8\sqrt{5}}{10} = -\frac{4\sqrt{5}}{5}$

3. 最終的な答え

(1) $8\sqrt{10}$

(2) $6\sqrt{7}$

(3) $18\sqrt{5}$

(4) $-16\sqrt{5}$

(5) $\frac{\sqrt{30}}{10}$

(6) $\frac{3\sqrt{10}}{8}$

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

(8) $-\frac{4\sqrt{5}}{5}$