画像に写っている数学の問題を解きます。画像には「Try」と「Exercise」という2つのセクションがあり、それぞれ複数の問題が含まれています。これらの問題を一つずつ解いていきます。

1. 問題の内容

画像に写っている数学の問題を解きます。画像には「Try」と「Exercise」という2つのセクションがあり、それぞれ複数の問題が含まれています。これらの問題を一つずつ解いていきます。

2. 解き方の手順

以下、画像に写っている問題を順番に解いていきます。

**Tryのセクション**

(1) $\sqrt{32} - 10 \times \sqrt{5}$:

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

$10 \times \sqrt{5}$ はそのまま。

よって、$4\sqrt{2} - 10\sqrt{5}$。

(2) $\frac{12}{\sqrt{3}} - \sqrt{6} \times \sqrt{18}$:

$\frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3}$

$\sqrt{6} \times \sqrt{18} = \sqrt{6 \times 18} = \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$

よって、$4\sqrt{3} - 6\sqrt{3} = -2\sqrt{3}$。

(3) $\frac{20}{\sqrt{5}} - 2\sqrt{15} \div \sqrt{3}$:

$\frac{20}{\sqrt{5}} = \frac{20\sqrt{5}}{5} = 4\sqrt{5}$

$2\sqrt{15} \div \sqrt{3} = 2 \sqrt{\frac{15}{3}} = 2\sqrt{5}$

よって、$4\sqrt{5} - 2\sqrt{5} = 2\sqrt{5}$。

(4) $(7\sqrt{3} - 2\sqrt{12}) \div \sqrt{3}$:

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$7\sqrt{3} - 2(2\sqrt{3}) = 7\sqrt{3} - 4\sqrt{3} = 3\sqrt{3}$

$(3\sqrt{3}) \div \sqrt{3} = 3$。

(5) $\sqrt{3}(\sqrt{2} + \sqrt{12})$:

$\sqrt{3} \times \sqrt{2} = \sqrt{6}$

$\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$

よって、$\sqrt{6} + 6$。

(6) $\sqrt{2}(\sqrt{2} - 3) - 3(\sqrt{2}+3)$:

$\sqrt{2}\times\sqrt{2} - 3\sqrt{2} - 3\sqrt{2} - 9 = 2 - 6\sqrt{2} - 9 = -7 - 6\sqrt{2}$

**Exerciseのセクション**

(1) $5\sqrt{3} - 3\sqrt{2} \times \sqrt{6}$:

$\sqrt{2} \times \sqrt{6} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$5\sqrt{3} - 3(2\sqrt{3}) = 5\sqrt{3} - 6\sqrt{3} = -\sqrt{3}$。

(2) $5\sqrt{2} - \sqrt{6} \times \sqrt{3}$:

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

$5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}$。

(3) $\sqrt{6} - \sqrt{12} \times \sqrt{8}$:

$\sqrt{12} \times \sqrt{8} = \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$

$\sqrt{6} - 4\sqrt{6} = -3\sqrt{6}$。

(4) $(\sqrt{12} - \sqrt{54}) \div \sqrt{2}$:

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$

$(2\sqrt{3} - 3\sqrt{6}) \div \sqrt{2} = \frac{2\sqrt{3}}{\sqrt{2}} - \frac{3\sqrt{6}}{\sqrt{2}} = 2\sqrt{\frac{3}{2}} - 3\sqrt{3} = \sqrt{6} - 3\sqrt{3}$.

(5) $\sqrt{18} + \sqrt{96} \div \sqrt{3}$:

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

$\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$

$\sqrt{96} \div \sqrt{3} = \sqrt{\frac{96}{3}} = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$

$3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}$

(6) $\sqrt{14} \div 3\sqrt{7} - \sqrt{18}$:

$\sqrt{14} \div 3\sqrt{7} = \frac{\sqrt{14}}{3\sqrt{7}} = \frac{\sqrt{2}}{3}$

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

$\frac{\sqrt{2}}{3} - 3\sqrt{2} = \frac{\sqrt{2} - 9\sqrt{2}}{3} = -\frac{8\sqrt{2}}{3}$

(7) $\frac{10}{\sqrt{5}} + \sqrt{15} \times \sqrt{3}$:

$\frac{10}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}$

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

$2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$

(8) $\frac{6}{\sqrt{3}} - 3\sqrt{6} \times \sqrt{8}$:

$\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

$\sqrt{6} \times \sqrt{8} = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

$2\sqrt{3} - 3(4\sqrt{3}) = 2\sqrt{3} - 12\sqrt{3} = -10\sqrt{3}$

(9) $3\sqrt{10} \times \sqrt{2} - \frac{15}{\sqrt{5}}$:

$3\sqrt{10} \times \sqrt{2} = 3\sqrt{20} = 3\sqrt{4 \times 5} = 3(2\sqrt{5}) = 6\sqrt{5}$

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

$6\sqrt{5} - 3\sqrt{5} = 3\sqrt{5}$

(10) $\frac{12}{\sqrt{3}} + \sqrt{72} \div \sqrt{6}$:

$\frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3}$

$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

$\sqrt{72} \div \sqrt{6} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$

(11) $\frac{\sqrt{8}}{4} - \sqrt{27} \div \sqrt{6}$:

$\sqrt{8} = 2\sqrt{2}$

$\sqrt{27} = 3\sqrt{3}$

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

(12) $\frac{14}{\sqrt{7}} + \sqrt{56} \div 2\sqrt{2}$:

$\frac{14}{\sqrt{7}} = \frac{14\sqrt{7}}{7} = 2\sqrt{7}$

$\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}$

$2\sqrt{14} \div 2\sqrt{2} = \sqrt{\frac{14}{2}} = \sqrt{7}$

$2\sqrt{7} + \sqrt{7} = 3\sqrt{7}$

(13) $(5\sqrt{2} - 2\sqrt{8}) \div \sqrt{2}$:

$\sqrt{8} = 2\sqrt{2}$

$5\sqrt{2} - 2(2\sqrt{2}) = 5\sqrt{2} - 4\sqrt{2} = \sqrt{2}$

$\frac{\sqrt{2}}{\sqrt{2}} = 1$

(14) $(\sqrt{50} - \sqrt{18}) \div \sqrt{2}$:

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

$(5\sqrt{2} - 3\sqrt{2}) \div \sqrt{2} = 2\sqrt{2} \div \sqrt{2} = 2$

(15) $(\sqrt{27} - 3\sqrt{12}) \div \sqrt{3}$:

$\sqrt{27} = 3\sqrt{3}$

$\sqrt{12} = 2\sqrt{3}$

$(3\sqrt{3} - 3(2\sqrt{3})) \div \sqrt{3} = (3\sqrt{3} - 6\sqrt{3}) \div \sqrt{3} = -3\sqrt{3} \div \sqrt{3} = -3$

(16) $\sqrt{6}(\sqrt{3} + \sqrt{2})$:

$\sqrt{6}\sqrt{3} + \sqrt{6}\sqrt{2} = \sqrt{18} + \sqrt{12} = 3\sqrt{2} + 2\sqrt{3}$

(17) $\sqrt{3}(2\sqrt{6} - \sqrt{12})$:

$2\sqrt{3}\sqrt{6} - \sqrt{3}\sqrt{12} = 2\sqrt{18} - \sqrt{36} = 2(3\sqrt{2}) - 6 = 6\sqrt{2} - 6$

(18) $\sqrt{2}(\sqrt{6} - \sqrt{8})$:

$\sqrt{2}\sqrt{6} - \sqrt{2}\sqrt{8} = \sqrt{12} - \sqrt{16} = 2\sqrt{3} - 4$

(19) $\sqrt{3}(2 + \sqrt{6}) - \sqrt{2}(3 - \sqrt{6})$:

$2\sqrt{3} + \sqrt{18} - 3\sqrt{2} + \sqrt{12} = 2\sqrt{3} + 3\sqrt{2} - 3\sqrt{2} + 2\sqrt{3} = 4\sqrt{3}$

(20) $3\sqrt{2}(\sqrt{6} - 1) + \sqrt{3}(2 + \sqrt{6})$:

$3\sqrt{12} - 3\sqrt{2} + 2\sqrt{3} + \sqrt{18} = 3(2\sqrt{3}) - 3\sqrt{2} + 2\sqrt{3} + 3\sqrt{2} = 6\sqrt{3} + 2\sqrt{3} = 8\sqrt{3}$

(21) $\sqrt{2}(3\sqrt{14} - 5\sqrt{2}) - \sqrt{3}(\sqrt{27} - \sqrt{21})$:

$3\sqrt{28} - 5\sqrt{4} - \sqrt{81} + \sqrt{63} = 3(2\sqrt{7}) - 10 - 9 + 3\sqrt{7} = 6\sqrt{7} - 10 - 9 + 3\sqrt{7} = 9\sqrt{7} - 19$

3. 最終的な答え

以下にそれぞれの問題の答えをまとめます。

**Try**

(1) $4\sqrt{2} - 10\sqrt{5}$

(2) $-2\sqrt{3}$

(3) $2\sqrt{5}$

(4) $3$

(5) $\sqrt{6} + 6$

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

**Exercise**

(1) $-\sqrt{3}$

(2) $2\sqrt{2}$

(3) $-3\sqrt{6}$

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

(5) $7\sqrt{2}$

(6) $-\frac{8\sqrt{2}}{3}$

(7) $5\sqrt{5}$

(8) $-10\sqrt{3}$

(9) $3\sqrt{5}$

(10) $6\sqrt{3}$

(11) $-\sqrt{2}$

(12) $3\sqrt{7}$

(13) $1$

(14) $2$

(15) $-3$

(16) $3\sqrt{2} + 2\sqrt{3}$

(17) $6\sqrt{2} - 6$

(18) $2\sqrt{3} - 4$

(19) $4\sqrt{3}$

(20) $8\sqrt{3}$

(21) $9\sqrt{7} - 19$