以下の8つの計算問題を解く必要があります。 (1) $\sqrt{18} \times \frac{1}{\sqrt{3}} \div \frac{1}{\sqrt{2}}$ (2) $\sqrt{125} - 4\sqrt{5} - \sqrt{45}$ (3) $\sqrt{3} \times 5\sqrt{6} - \frac{12}{\sqrt{2}}$ (4) $\sqrt{2}(\sqrt{24} - \sqrt{6})$ (5) $(2\sqrt{3} - 1)^2$ (6) $(\sqrt{2} - \sqrt{3})(2\sqrt{2} + 5\sqrt{3})$ (7) $(\sqrt{2} - 3)^2 + 6\sqrt{2}$ (8) $(\sqrt{3} + 1)(\sqrt{3} + 2) - \frac{9}{\sqrt{3}}$

はい、承知いたしました。画像にある数学の問題を解いていきます。

1. 問題の内容

以下の8つの計算問題を解く必要があります。

(1) $\sqrt{18} \times \frac{1}{\sqrt{3}} \div \frac{1}{\sqrt{2}}$

(2) $\sqrt{125} - 4\sqrt{5} - \sqrt{45}$

(3) $\sqrt{3} \times 5\sqrt{6} - \frac{12}{\sqrt{2}}$

(4) $\sqrt{2}(\sqrt{24} - \sqrt{6})$

(5) $(2\sqrt{3} - 1)^2$

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

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

(8) $(\sqrt{3} + 1)(\sqrt{3} + 2) - \frac{9}{\sqrt{3}}$

2. 解き方の手順

各問題ごとに手順を説明します。

(1) $\sqrt{18} \times \frac{1}{\sqrt{3}} \div \frac{1}{\sqrt{2}} = \sqrt{18} \times \frac{1}{\sqrt{3}} \times \sqrt{2} = \sqrt{\frac{18 \times 2}{3}} = \sqrt{12} = 2\sqrt{3}$

(2) $\sqrt{125} - 4\sqrt{5} - \sqrt{45} = \sqrt{25 \times 5} - 4\sqrt{5} - \sqrt{9 \times 5} = 5\sqrt{5} - 4\sqrt{5} - 3\sqrt{5} = (5 - 4 - 3)\sqrt{5} = -2\sqrt{5}$

(3) $\sqrt{3} \times 5\sqrt{6} - \frac{12}{\sqrt{2}} = 5\sqrt{18} - \frac{12\sqrt{2}}{2} = 5\sqrt{9 \times 2} - 6\sqrt{2} = 5 \times 3\sqrt{2} - 6\sqrt{2} = 15\sqrt{2} - 6\sqrt{2} = 9\sqrt{2}$

(4) $\sqrt{2}(\sqrt{24} - \sqrt{6}) = \sqrt{2}(\sqrt{4 \times 6} - \sqrt{6}) = \sqrt{2}(2\sqrt{6} - \sqrt{6}) = \sqrt{2}\sqrt{6} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

(5) $(2\sqrt{3} - 1)^2 = (2\sqrt{3})^2 - 2(2\sqrt{3})(1) + 1^2 = 4 \times 3 - 4\sqrt{3} + 1 = 12 - 4\sqrt{3} + 1 = 13 - 4\sqrt{3}$

(6) $(\sqrt{2} - \sqrt{3})(2\sqrt{2} + 5\sqrt{3}) = \sqrt{2}(2\sqrt{2} + 5\sqrt{3}) - \sqrt{3}(2\sqrt{2} + 5\sqrt{3}) = 4 + 5\sqrt{6} - 2\sqrt{6} - 15 = -11 + 3\sqrt{6}$

(7) $(\sqrt{2} - 3)^2 + 6\sqrt{2} = (\sqrt{2})^2 - 2(\sqrt{2})(3) + 3^2 + 6\sqrt{2} = 2 - 6\sqrt{2} + 9 + 6\sqrt{2} = 11$

(8) $(\sqrt{3} + 1)(\sqrt{3} + 2) - \frac{9}{\sqrt{3}} = 3 + 2\sqrt{3} + \sqrt{3} + 2 - \frac{9\sqrt{3}}{3} = 5 + 3\sqrt{3} - 3\sqrt{3} = 5$

3. 最終的な答え

(1) $2\sqrt{3}$

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

(3) $9\sqrt{2}$

(4) $2\sqrt{3}$

(5) $13 - 4\sqrt{3}$

(6) $-11 + 3\sqrt{6}$

(7) $11$

(8) $5$