与えられた複数の数式を計算し、答えを求める問題です。

1. 問題の内容

与えられた複数の数式を計算し、答えを求める問題です。

2. 解き方の手順

それぞれの問題ごとに、分配法則や展開、有理化などの計算テクニックを用いて計算を進めます。

(1) $(\sqrt{2}+1)(\sqrt{3}+2) = \sqrt{6} + 2\sqrt{2} + \sqrt{3} + 2$

(2) $(3\sqrt{2}-1)(\sqrt{2}-3) = 3\cdot 2 - 9\sqrt{2} - \sqrt{2} + 3 = 6 - 10\sqrt{2} + 3 = 9 - 10\sqrt{2}$

(3) $(\sqrt{2}-3)(\sqrt{2}+5) = 2 + 5\sqrt{2} - 3\sqrt{2} - 15 = 2\sqrt{2} - 13$

(4) $(2\sqrt{3}+6)(2\sqrt{3}-7) = 4\cdot 3 - 14\sqrt{3} + 12\sqrt{3} - 42 = 12 - 2\sqrt{3} - 42 = -30 - 2\sqrt{3}$

(5) $(\sqrt{6}-2\sqrt{3})(\sqrt{6}+\sqrt{3}) = 6 + \sqrt{18} - 2\sqrt{18} - 2\cdot 3 = 6 + 3\sqrt{2} - 6\sqrt{2} - 6 = -3\sqrt{2}$

(6) $(4\sqrt{3}-2\sqrt{6})(4\sqrt{3}+\sqrt{6}) = 16\cdot 3 + 4\sqrt{18} - 8\sqrt{18} - 2\cdot 6 = 48 + 12\sqrt{2} - 24\sqrt{2} - 12 = 36 - 12\sqrt{2}$

(7) $(\sqrt{2}+1)^2 = (\sqrt{2})^2 + 2\sqrt{2} + 1 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2}$

(8) $(2\sqrt{3}+\sqrt{5})^2 = (2\sqrt{3})^2 + 2\cdot 2\sqrt{3}\sqrt{5} + (\sqrt{5})^2 = 4\cdot 3 + 4\sqrt{15} + 5 = 12 + 4\sqrt{15} + 5 = 17 + 4\sqrt{15}$

(9) $(\sqrt{3}+\sqrt{6})^2 = (\sqrt{3})^2 + 2\sqrt{18} + (\sqrt{6})^2 = 3 + 2\cdot 3\sqrt{2} + 6 = 9 + 6\sqrt{2}$

(10) $(\sqrt{3}-4)^2 = (\sqrt{3})^2 - 2\cdot 4\sqrt{3} + 4^2 = 3 - 8\sqrt{3} + 16 = 19 - 8\sqrt{3}$

(11) $(\sqrt{5}-\sqrt{2})^2 = (\sqrt{5})^2 - 2\sqrt{10} + (\sqrt{2})^2 = 5 - 2\sqrt{10} + 2 = 7 - 2\sqrt{10}$

(12) $(2\sqrt{3}-\sqrt{2})^2 = (2\sqrt{3})^2 - 2\cdot 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2 = 4\cdot 3 - 4\sqrt{6} + 2 = 12 - 4\sqrt{6} + 2 = 14 - 4\sqrt{6}$

(13) $(4+2\sqrt{3})(4-2\sqrt{3}) = 4^2 - (2\sqrt{3})^2 = 16 - 4\cdot 3 = 16 - 12 = 4$

(14) $(2\sqrt{3}+2\sqrt{2})(2\sqrt{3}-2\sqrt{2}) = (2\sqrt{3})^2 - (2\sqrt{2})^2 = 4\cdot 3 - 4\cdot 2 = 12 - 8 = 4$

(15) $(\sqrt{3}+2)^2 - \sqrt{48} = 3 + 4\sqrt{3} + 4 - 4\sqrt{3} = 7$

(16) $(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) + \frac{\sqrt{48}}{\sqrt{12}} = (5-3) + \sqrt{\frac{48}{12}} = 2 + \sqrt{4} = 2 + 2 = 4$

(17) $(3\sqrt{3}+1)(1-\sqrt{3}) - (1-\sqrt{3})^2 = 3\sqrt{3} - 9 + 1 - \sqrt{3} - (1 - 2\sqrt{3} + 3) = 2\sqrt{3} - 8 - (4 - 2\sqrt{3}) = 2\sqrt{3} - 8 - 4 + 2\sqrt{3} = 4\sqrt{3} - 12$

(18) $(4-\sqrt{2})^2 - (\sqrt{5}-2\sqrt{3})(\sqrt{5}+2\sqrt{3}) = (16 - 8\sqrt{2} + 2) - (5 - 4\cdot 3) = 18 - 8\sqrt{2} - (5 - 12) = 18 - 8\sqrt{2} - (-7) = 18 - 8\sqrt{2} + 7 = 25 - 8\sqrt{2}$

3. 最終的な答え

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

(2) $9 - 10\sqrt{2}$

(3) $2\sqrt{2} - 13$

(4) $-30 - 2\sqrt{3}$

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

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

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

(8) $17 + 4\sqrt{15}$

(9) $9 + 6\sqrt{2}$

(10) $19 - 8\sqrt{3}$

(11) $7 - 2\sqrt{10}$

(12) $14 - 4\sqrt{6}$

(13) $4$

(14) $4$

(15) $7$

(16) $4$

(17) $4\sqrt{3} - 12$

(18) $25 - 8\sqrt{2}$