与えられた計算問題を解く。問題はStep2の(1)から(6)とStep3の(1)から(4)まで、合計10問ある。

1. 問題の内容

与えられた計算問題を解く。問題はStep2の(1)から(6)とStep3の(1)から(4)まで、合計10問ある。

2. 解き方の手順

Step2

(1) $(\sqrt{5}-2)^2 = (\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot 2 + 2^2 = 5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5}$

(2) $(2\sqrt{6}+6)^2 = (2\sqrt{6})^2 + 2 \cdot 2\sqrt{6} \cdot 6 + 6^2 = 4 \cdot 6 + 24\sqrt{6} + 36 = 24 + 24\sqrt{6} + 36 = 60 + 24\sqrt{6}$

(3) $(3\sqrt{3}-1)^2 = (3\sqrt{3})^2 - 2 \cdot 3\sqrt{3} \cdot 1 + 1^2 = 9 \cdot 3 - 6\sqrt{3} + 1 = 27 - 6\sqrt{3} + 1 = 28 - 6\sqrt{3}$

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

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

(6) $\frac{4}{\sqrt{11}-3} = \frac{4(\sqrt{11}+3)}{(\sqrt{11}-3)(\sqrt{11}+3)} = \frac{4(\sqrt{11}+3)}{11-9} = \frac{4(\sqrt{11}+3)}{2} = 2(\sqrt{11}+3) = 2\sqrt{11}+6$

Step3

(1) $(2\sqrt{2}+\sqrt{3})^2 = (2\sqrt{2})^2 + 2 \cdot 2\sqrt{2} \cdot \sqrt{3} + (\sqrt{3})^2 = 4 \cdot 2 + 4\sqrt{6} + 3 = 8 + 4\sqrt{6} + 3 = 11 + 4\sqrt{6}$

(2) $(5\sqrt{6}-2\sqrt{5})^2 = (5\sqrt{6})^2 - 2 \cdot 5\sqrt{6} \cdot 2\sqrt{5} + (2\sqrt{5})^2 = 25 \cdot 6 - 20\sqrt{30} + 4 \cdot 5 = 150 - 20\sqrt{30} + 20 = 170 - 20\sqrt{30}$

(3) $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} = \frac{(\sqrt{5}-\sqrt{3})(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{(\sqrt{5}-\sqrt{3})^2}{5-3} = \frac{(\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot \sqrt{3} + (\sqrt{3})^2}{2} = \frac{5 - 2\sqrt{15} + 3}{2} = \frac{8 - 2\sqrt{15}}{2} = 4 - \sqrt{15}$

(4) $\frac{5\sqrt{6}-2\sqrt{5}}{5\sqrt{6}+2\sqrt{5}} = \frac{(5\sqrt{6}-2\sqrt{5})(5\sqrt{6}-2\sqrt{5})}{(5\sqrt{6}+2\sqrt{5})(5\sqrt{6}-2\sqrt{5})} = \frac{(5\sqrt{6}-2\sqrt{5})^2}{(5\sqrt{6})^2-(2\sqrt{5})^2} = \frac{(5\sqrt{6})^2 - 2 \cdot 5\sqrt{6} \cdot 2\sqrt{5} + (2\sqrt{5})^2}{25 \cdot 6 - 4 \cdot 5} = \frac{25 \cdot 6 - 20\sqrt{30} + 4 \cdot 5}{150 - 20} = \frac{150 - 20\sqrt{30} + 20}{130} = \frac{170 - 20\sqrt{30}}{130} = \frac{17 - 2\sqrt{30}}{13}$

3. 最終的な答え

Step2

(1) $9 - 4\sqrt{5}$

(2) $60 + 24\sqrt{6}$

(3) $28 - 6\sqrt{3}$

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

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

(6) $2\sqrt{11}+6$

Step3

(1) $11 + 4\sqrt{6}$

(2) $170 - 20\sqrt{30}$

(3) $4 - \sqrt{15}$

(4) $\frac{17 - 2\sqrt{30}}{13}$