与えられた8つの数式を計算し、簡略化します。

1. 問題の内容

与えられた8つの数式を計算し、簡略化します。

2. 解き方の手順

(1) $-2\sqrt{2} + \sqrt{18}$

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$なので、

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

(2) $\sqrt{27} + \sqrt{12} - \sqrt{3}$

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

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$3\sqrt{3} + 2\sqrt{3} - \sqrt{3} = 4\sqrt{3}$

(3) $\sqrt{125} + \sqrt{80} - \sqrt{45}$

$\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$

$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$

$5\sqrt{5} + 4\sqrt{5} - 3\sqrt{5} = 6\sqrt{5}$

(4) $\sqrt{6} \times \sqrt{8} - \sqrt{15} \div \sqrt{5}$

$\sqrt{6} \times \sqrt{8} = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$

$\sqrt{15} \div \sqrt{5} = \sqrt{3}$

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

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

$(\sqrt{5} + 4)^2 = (\sqrt{5})^2 + 2 \times 4 \times \sqrt{5} + 4^2 = 5 + 8\sqrt{5} + 16 = 21 + 8\sqrt{5}$

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

$(\sqrt{3} + 1)(\sqrt{3} + 2) = (\sqrt{3})^2 + 2\sqrt{3} + \sqrt{3} + 2 = 3 + 3\sqrt{3} + 2 = 5 + 3\sqrt{3}$

(7) $\sqrt{5}(\sqrt{5} - 3) + \sqrt{20}$

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

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

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

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

$(\sqrt{2} + \sqrt{5})(\sqrt{5} - 2\sqrt{2}) = \sqrt{2}\sqrt{5} - 2(\sqrt{2})^2 + (\sqrt{5})^2 - 2\sqrt{5}\sqrt{2} = \sqrt{10} - 4 + 5 - 2\sqrt{10} = 1 - \sqrt{10}$

3. 最終的な答え

(1) $\sqrt{2}$

(2) $4\sqrt{3}$

(3) $6\sqrt{5}$

(4) $3\sqrt{3}$

(5) $21 + 8\sqrt{5}$

(6) $5 + 3\sqrt{3}$

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

(8) $1 - \sqrt{10}$