与えられた9個の数式をそれぞれ計算し、簡略化します。数式は以下の通りです。 (1) $5\sqrt{2} \times 3\sqrt{6}$ (2) $\sqrt{3}\sqrt{48}$ (3) $\frac{\sqrt{42}}{\sqrt{7}}$ (4) $\sqrt{3}(2\sqrt{3} - \sqrt{6})$ (5) $\sqrt{5}(3\sqrt{20} - 4\sqrt{45})$ (6) $(\sqrt{3} + \sqrt{7})^2$ (7) $(\sqrt{6} - 3\sqrt{2})^2$ (8) $(2\sqrt{2} - \sqrt{3})(2\sqrt{2} + \sqrt{3})$ (9) $(\sqrt{20} + \sqrt{3})(\sqrt{5} - \sqrt{27})$

1. 問題の内容

与えられた9個の数式をそれぞれ計算し、簡略化します。数式は以下の通りです。

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

(2) $\sqrt{3}\sqrt{48}$

(3) $\frac{\sqrt{42}}{\sqrt{7}}$

(4) $\sqrt{3}(2\sqrt{3} - \sqrt{6})$

(5) $\sqrt{5}(3\sqrt{20} - 4\sqrt{45})$

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

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

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

(9) $(\sqrt{20} + \sqrt{3})(\sqrt{5} - \sqrt{27})$

2. 解き方の手順

それぞれの数式を以下のように計算します。

(1) $5\sqrt{2} \times 3\sqrt{6} = 15\sqrt{12} = 15 \times 2\sqrt{3} = 30\sqrt{3}$

(2) $\sqrt{3}\sqrt{48} = \sqrt{3 \times 48} = \sqrt{144} = 12$

(3) $\frac{\sqrt{42}}{\sqrt{7}} = \sqrt{\frac{42}{7}} = \sqrt{6}$

(4) $\sqrt{3}(2\sqrt{3} - \sqrt{6}) = 2\sqrt{3}\sqrt{3} - \sqrt{3}\sqrt{6} = 2 \times 3 - \sqrt{18} = 6 - 3\sqrt{2}$

(5) $\sqrt{5}(3\sqrt{20} - 4\sqrt{45}) = \sqrt{5}(3 \times 2\sqrt{5} - 4 \times 3\sqrt{5}) = \sqrt{5}(6\sqrt{5} - 12\sqrt{5}) = \sqrt{5}(-6\sqrt{5}) = -6 \times 5 = -30$

(6) $(\sqrt{3} + \sqrt{7})^2 = (\sqrt{3})^2 + 2\sqrt{3}\sqrt{7} + (\sqrt{7})^2 = 3 + 2\sqrt{21} + 7 = 10 + 2\sqrt{21}$

(7) $(\sqrt{6} - 3\sqrt{2})^2 = (\sqrt{6})^2 - 2(\sqrt{6})(3\sqrt{2}) + (3\sqrt{2})^2 = 6 - 6\sqrt{12} + 9 \times 2 = 6 - 6 \times 2\sqrt{3} + 18 = 24 - 12\sqrt{3}$

(8) $(2\sqrt{2} - \sqrt{3})(2\sqrt{2} + \sqrt{3}) = (2\sqrt{2})^2 - (\sqrt{3})^2 = 4 \times 2 - 3 = 8 - 3 = 5$

(9) $(\sqrt{20} + \sqrt{3})(\sqrt{5} - \sqrt{27}) = (2\sqrt{5} + \sqrt{3})(\sqrt{5} - 3\sqrt{3}) = 2\sqrt{5}\sqrt{5} - 6\sqrt{5}\sqrt{3} + \sqrt{3}\sqrt{5} - 3\sqrt{3}\sqrt{3} = 2 \times 5 - 6\sqrt{15} + \sqrt{15} - 3 \times 3 = 10 - 5\sqrt{15} - 9 = 1 - 5\sqrt{15}$

3. 最終的な答え

(1) $30\sqrt{3}$

(2) $12$

(3) $\sqrt{6}$

(4) $6 - 3\sqrt{2}$

(5) $-30$

(6) $10 + 2\sqrt{21}$

(7) $24 - 12\sqrt{3}$

(8) $5$

(9) $1 - 5\sqrt{15}$