与えられた計算問題を解きます。具体的には、以下の12個の計算問題を解きます。 (3) $(2\sqrt{2}-1)(2\sqrt{2}-3)$ (4) $(\sqrt{5}+\sqrt{2})^2$ (5) $(\sqrt{2}-1)^2$ (6) $(1-\sqrt{7})^2$ (7) $(\sqrt{6}-\sqrt{2})^2$ (8) $(\sqrt{7}+2)(\sqrt{7}-2)$ (9) $(3+\sqrt{5})(3-\sqrt{5})$ (10) $(\sqrt{2}+\sqrt{6})(\sqrt{2}-\sqrt{6})$ (11) $(2+\sqrt{2})(\sqrt{2}-1)$ (12) $(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})$

1. 問題の内容

与えられた計算問題を解きます。具体的には、以下の12個の計算問題を解きます。

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

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

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

(6) $(1-\sqrt{7})^2$

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

(8) $(\sqrt{7}+2)(\sqrt{7}-2)$

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

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

(11) $(2+\sqrt{2})(\sqrt{2}-1)$

(12) $(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})$

2. 解き方の手順

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

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

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

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

$= 11 - 8\sqrt{2}$

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

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

$= 5 + 2\sqrt{10} + 2$

$= 7 + 2\sqrt{10}$

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

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

$= 2 - 2\sqrt{2} + 1$

$= 3 - 2\sqrt{2}$

(6) $(1-\sqrt{7})^2$

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

$= 1 - 2\sqrt{7} + 7$

$= 8 - 2\sqrt{7}$

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

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

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

$= 8 - 2\sqrt{4\cdot3}$

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

$= 8 - 4\sqrt{3}$

(8) $(\sqrt{7}+2)(\sqrt{7}-2)$

$= (\sqrt{7})^2 - 2^2$

$= 7 - 4$

$= 3$

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

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

$= 9 - 5$

$= 4$

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

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

$= 2 - 6$

$= -4$

(11) $(2+\sqrt{2})(\sqrt{2}-1)$

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

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

$= \sqrt{2}$

(12) $(\sqrt{12}-\sqrt{8})(\sqrt{3}+\sqrt{2})$

$= (\sqrt{4\cdot3}-\sqrt{4\cdot2})(\sqrt{3}+\sqrt{2})$

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

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

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

$= 2(3-2)$

$= 2(1)$

$= 2$

3. 最終的な答え

(3) $11 - 8\sqrt{2}$

(4) $7 + 2\sqrt{10}$

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

(6) $8 - 2\sqrt{7}$

(7) $8 - 4\sqrt{3}$

(8) $3$

(9) $4$

(10) $-4$

(11) $\sqrt{2}$

(12) $2$