与えられた複数の平方根の計算問題を解きます。具体的には、以下の6つの問題を解きます。 (1) $\sqrt{50} + \sqrt{32}$ (2) $\sqrt{27} - \sqrt{3}$ (3) $\sqrt{18} + \sqrt{8} - \sqrt{72}$ (4) $\sqrt{20} - \sqrt{80} - \sqrt{45}$ (5) $\sqrt{12} + \sqrt{18} - \sqrt{8}$ (6) $\sqrt{98} - \sqrt{27} + \sqrt{48} - \sqrt{32}$

1. 問題の内容

与えられた複数の平方根の計算問題を解きます。具体的には、以下の6つの問題を解きます。

(1) $\sqrt{50} + \sqrt{32}$

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

(3) $\sqrt{18} + \sqrt{8} - \sqrt{72}$

(4) $\sqrt{20} - \sqrt{80} - \sqrt{45}$

(5) $\sqrt{12} + \sqrt{18} - \sqrt{8}$

(6) $\sqrt{98} - \sqrt{27} + \sqrt{48} - \sqrt{32}$

2. 解き方の手順

各平方根を最も簡単な形に変形し、同類項をまとめます。

(1) $\sqrt{50} + \sqrt{32}$

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

$\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$

$5\sqrt{2} + 4\sqrt{2} = (5+4)\sqrt{2} = 9\sqrt{2}$

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

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

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

(3) $\sqrt{18} + \sqrt{8} - \sqrt{72}$

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

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

$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

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

(4) $\sqrt{20} - \sqrt{80} - \sqrt{45}$

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

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

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

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

(5) $\sqrt{12} + \sqrt{18} - \sqrt{8}$

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

$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

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

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

(6) $\sqrt{98} - \sqrt{27} + \sqrt{48} - \sqrt{32}$

$\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}$

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

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

$\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$

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

3. 最終的な答え

(1) $9\sqrt{2}$

(2) $2\sqrt{3}$

(3) $-\sqrt{2}$

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

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

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