以下の6つの式を計算します。 (1) $\sqrt{3} + \sqrt{48} - \sqrt{27}$ (2) $\sqrt{8} + \sqrt{50}$ (3) $(\sqrt{3} + \sqrt{5})^2$ (4) $(\sqrt{2} - \sqrt{3})^2$ (5) $(2\sqrt{3} - \sqrt{5})(3\sqrt{3} + 4\sqrt{5})$ (6) $(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})$

はい、承知いたしました。与えられた数学の問題を解きます。

1. 問題の内容

以下の6つの式を計算します。

(1) $\sqrt{3} + \sqrt{48} - \sqrt{27}$

(2) $\sqrt{8} + \sqrt{50}$

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

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

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

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

2. 解き方の手順

(1) $\sqrt{3} + \sqrt{48} - \sqrt{27}$

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

よって、$\sqrt{3} + 4\sqrt{3} - 3\sqrt{3} = 2\sqrt{3}$

(2) $\sqrt{8} + \sqrt{50}$

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

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

よって、$2\sqrt{2} + 5\sqrt{2} = 7\sqrt{2}$

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

$(\sqrt{3} + \sqrt{5})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{5}) + (\sqrt{5})^2 = 3 + 2\sqrt{15} + 5 = 8 + 2\sqrt{15}$

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

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

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

$(2\sqrt{3} - \sqrt{5})(3\sqrt{3} + 4\sqrt{5}) = (2\sqrt{3})(3\sqrt{3}) + (2\sqrt{3})(4\sqrt{5}) - (\sqrt{5})(3\sqrt{3}) - (\sqrt{5})(4\sqrt{5}) = 6(3) + 8\sqrt{15} - 3\sqrt{15} - 4(5) = 18 + 5\sqrt{15} - 20 = -2 + 5\sqrt{15}$

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

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

3. 最終的な答え

(1) $2\sqrt{3}$

(2) $7\sqrt{2}$

(3) $8 + 2\sqrt{15}$

(4) $5 - 2\sqrt{6}$

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

(6) $5$