与えられた数式を計算し、簡単にしてください。 問題は以下の通りです。 (52) (1) $(1-\sqrt{2})(3+4\sqrt{2})$ (2) $(4+5\sqrt{3})(4-\sqrt{3})$ (3) $(\sqrt{5}+\sqrt{2})^2$ (4) $(2\sqrt{3}-\sqrt{2})^2$ (5) $(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})$ (53) (1) $(6+\sqrt{2})(1+2\sqrt{2})$ (2) $(4-\sqrt{7})(5-2\sqrt{7})$ (3) $(1+2\sqrt{6})^2$ (4) $(3\sqrt{2}-2\sqrt{3})^2$ (5) $(2\sqrt{2}+3\sqrt{3})(2\sqrt{2}-3\sqrt{3})$

1. 問題の内容

与えられた数式を計算し、簡単にしてください。

問題は以下の通りです。

(52)

(1) $(1-\sqrt{2})(3+4\sqrt{2})$

(2) $(4+5\sqrt{3})(4-\sqrt{3})$

(3) $(\sqrt{5}+\sqrt{2})^2$

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

(5) $(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})$

(53)

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

(2) $(4-\sqrt{7})(5-2\sqrt{7})$

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

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

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

2. 解き方の手順

各問題ごとに解き方を説明します。

(52)

(1) $(1-\sqrt{2})(3+4\sqrt{2}) = 3 + 4\sqrt{2} - 3\sqrt{2} - 4(2) = 3 + \sqrt{2} - 8 = -5+\sqrt{2}$

(2) $(4+5\sqrt{3})(4-\sqrt{3}) = 16 - 4\sqrt{3} + 20\sqrt{3} - 5(3) = 16 + 16\sqrt{3} - 15 = 1 + 16\sqrt{3}$

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

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

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

(53)

(1) $(6+\sqrt{2})(1+2\sqrt{2}) = 6 + 12\sqrt{2} + \sqrt{2} + 2(2) = 6 + 13\sqrt{2} + 4 = 10 + 13\sqrt{2}$

(2) $(4-\sqrt{7})(5-2\sqrt{7}) = 20 - 8\sqrt{7} - 5\sqrt{7} + 2(7) = 20 - 13\sqrt{7} + 14 = 34 - 13\sqrt{7}$

(3) $(1+2\sqrt{6})^2 = 1^2 + 2(1)(2\sqrt{6}) + (2\sqrt{6})^2 = 1 + 4\sqrt{6} + 4(6) = 1 + 4\sqrt{6} + 24 = 25 + 4\sqrt{6}$

(4) $(3\sqrt{2}-2\sqrt{3})^2 = (3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 9(2) - 12\sqrt{6} + 4(3) = 18 - 12\sqrt{6} + 12 = 30 - 12\sqrt{6}$

(5) $(2\sqrt{2}+3\sqrt{3})(2\sqrt{2}-3\sqrt{3}) = (2\sqrt{2})^2 - (3\sqrt{3})^2 = 4(2) - 9(3) = 8 - 27 = -19$

3. 最終的な答え

(52)

(1) $-5+\sqrt{2}$

(2) $1 + 16\sqrt{3}$

(3) $7 + 2\sqrt{10}$

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

(5) $4$

(53)

(1) $10 + 13\sqrt{2}$

(2) $34 - 13\sqrt{7}$

(3) $25 + 4\sqrt{6}$

(4) $30 - 12\sqrt{6}$

(5) $-19$