次の6つの式を計算します。 (1) $4\sqrt{3} \times 5\sqrt{7}$ (2) $\sqrt{2} \times \sqrt{6} \times \sqrt{18}$ (3) $(3\sqrt{5}+2\sqrt{3})(4\sqrt{5}-\sqrt{3})$ (4) $(\sqrt{5}+2)^2$ (5) $(3\sqrt{2}-\sqrt{6})^2$ (6) $(2\sqrt{2}+\sqrt{3})(2\sqrt{2}-\sqrt{3})$

はい、承知いたしました。画像の数学の問題を解きます。

1. 問題の内容

次の6つの式を計算します。

(1) $4\sqrt{3} \times 5\sqrt{7}$

(2) $\sqrt{2} \times \sqrt{6} \times \sqrt{18}$

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

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

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

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

2. 解き方の手順

(1) $4\sqrt{3} \times 5\sqrt{7} = 4 \times 5 \times \sqrt{3} \times \sqrt{7} = 20\sqrt{21}$

(2) $\sqrt{2} \times \sqrt{6} \times \sqrt{18} = \sqrt{2 \times 6 \times 18} = \sqrt{2 \times 2 \times 3 \times 2 \times 3 \times 3} = \sqrt{2^3 \times 3^3} = \sqrt{2^2 \times 2 \times 3^2 \times 3} = 2 \times 3 \sqrt{2 \times 3} = 6\sqrt{6} \times \sqrt{2 \times 3 \times 3} = \sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}$

別解: $\sqrt{2} \times \sqrt{6} \times \sqrt{18} = \sqrt{2} \times \sqrt{2} \sqrt{3} \times 3\sqrt{2} = 2 \times \sqrt{3} \times 3 \sqrt{2} = 6\sqrt{6}$

(3) $(3\sqrt{5}+2\sqrt{3})(4\sqrt{5}-\sqrt{3}) = 3\sqrt{5} \times 4\sqrt{5} + 3\sqrt{5} \times (-\sqrt{3}) + 2\sqrt{3} \times 4\sqrt{5} + 2\sqrt{3} \times (-\sqrt{3}) = 12 \times 5 - 3\sqrt{15} + 8\sqrt{15} - 2 \times 3 = 60 + 5\sqrt{15} - 6 = 54 + 5\sqrt{15}$

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

(5) $(3\sqrt{2}-\sqrt{6})^2 = (3\sqrt{2})^2 - 2 \times 3\sqrt{2} \times \sqrt{6} + (\sqrt{6})^2 = 9 \times 2 - 6\sqrt{12} + 6 = 18 - 6 \times 2\sqrt{3} + 6 = 24 - 12\sqrt{3}$

(6) $(2\sqrt{2}+\sqrt{3})(2\sqrt{2}-\sqrt{3}) = (2\sqrt{2})^2 - (\sqrt{3})^2 = 4 \times 2 - 3 = 8 - 3 = 5$

3. 最終的な答え

(1) $20\sqrt{21}$

(2) $6\sqrt{6}$

(3) $54 + 5\sqrt{15}$

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

(5) $24 - 12\sqrt{3}$

(6) $5$