次の4つの式を計算せよ。 (1) $(\sqrt{10} - 2\sqrt{2})(\sqrt{5} + 2)$ (2) $(\sqrt{3} - \sqrt{2})(2\sqrt{3} + 5\sqrt{2})$ (3) $(\sqrt{2} - \sqrt{6})^2$ (4) $(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2})$

1. 問題の内容

次の4つの式を計算せよ。

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

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

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

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

2. 解き方の手順

(1)

$(\sqrt{10} - 2\sqrt{2})(\sqrt{5} + 2) = \sqrt{10} \times \sqrt{5} + \sqrt{10} \times 2 - 2\sqrt{2} \times \sqrt{5} - 2\sqrt{2} \times 2$

$= \sqrt{50} + 2\sqrt{10} - 2\sqrt{10} - 4\sqrt{2}$

$= \sqrt{25 \times 2} + 2\sqrt{10} - 2\sqrt{10} - 4\sqrt{2}$

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

$= \sqrt{2}$

(2)

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

$= 2 \times 3 + 5\sqrt{6} - 2\sqrt{6} - 5 \times 2$

$= 6 + 3\sqrt{6} - 10$

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

(3)

$(\sqrt{2} - \sqrt{6})^2 = (\sqrt{2})^2 - 2 \times \sqrt{2} \times \sqrt{6} + (\sqrt{6})^2$

$= 2 - 2\sqrt{12} + 6$

$= 8 - 2\sqrt{4 \times 3}$

$= 8 - 2 \times 2\sqrt{3}$

$= 8 - 4\sqrt{3}$

(4)

$(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2})$ は $(a - b)(a + b) = a^2 - b^2$ の形なので、

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

$= 4 \times 3 - 2$

$= 12 - 2$

$= 10$

3. 最終的な答え

(1) $\sqrt{2}$

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

(3) $8 - 4\sqrt{3}$

(4) $10$