解説
^2=(\sqrt{5})^2%2B 2\sqrt{5}\sqrt{2}%2B (\sqrt{2})^2) \( \displaystyle (\sqrt{5}+ \sqrt{2})^2=(\sqrt{5})^2+ 2\sqrt{5}\sqrt{2}+ (\sqrt{2})^2 \)
 \( \displaystyle =5+ 2\sqrt{10}+ 2=7+ 2\sqrt{10} \)
(\sqrt{5}-1)=(\sqrt{5})^2-1^2) \( \displaystyle (\sqrt{5}+ 1)(\sqrt{5}-1)=(\sqrt{5})^2-1^2 \)
 \( \displaystyle =5-1=4 \)
^2=(\sqrt{5})^2-2\times \sqrt{5}\times \sqrt{3}%2B (\sqrt{3})^2) \( \displaystyle (\sqrt{5}- \sqrt{3})^2=(\sqrt{5})^2-2\times \sqrt{5}\times \sqrt{3}+ (\sqrt{3})^2 \)
 \( \displaystyle =5-2\sqrt{15}+ 3=8-2\sqrt{15} \)
(x+1)(x+2)=x2+(1+2)x+(1×2)と同様にして
(\sqrt{5}%2B 2)=(\sqrt{5})^2%2B (1%2B 2)\sqrt{5}%2B (1\times 2)) \( \displaystyle (\sqrt{5}+ 1)(\sqrt{5}+ 2)=(\sqrt{5})^2+ (1+ 2)\sqrt{5}+ (1\times 2) \)
 \( \displaystyle =5 + 3\sqrt{5} + 2=7 + 3\sqrt{5} \)
 \( \displaystyle \sqrt{45}- \sqrt{20}=\sqrt{3^2\times 5}- \sqrt{2^2\times 5} \)
 \( \displaystyle =3\sqrt{5}- 2\sqrt{5}=\sqrt{5} \)
 \( \displaystyle \sqrt{20}+ 6\sqrt{5}=\sqrt{2^2\times 5}+ 6\sqrt{5} \)
 \( \displaystyle =2\sqrt{5}+ 6\sqrt{5}=8\sqrt{5} \)
^2=(\sqrt{3})^2%2B 2\times \sqrt{3}\times \sqrt{2}%2B (\sqrt{2})^2) \( \displaystyle (\sqrt{3}+ \sqrt{2})^2=(\sqrt{3})^2+ 2\times \sqrt{3}\times \sqrt{2}+ (\sqrt{2})^2 \)
 \( \displaystyle =3+ 2\sqrt{6}+ 2=5+ 2\sqrt{6} \)
(2 %2B \sqrt{3})=2^2-(\sqrt{3})^2) \( \displaystyle (2- \sqrt{3})(2 + \sqrt{3})=2^2-(\sqrt{3})^2 \)
 \( \displaystyle =4-3=1 \)
^2=(\sqrt{3})^2 - 2\times \sqrt{3}\times \sqrt{2}%2B (\sqrt{2})^2) \( \displaystyle (\sqrt{3}- \sqrt{2})^2=(\sqrt{3})^2 - 2\times \sqrt{3}\times \sqrt{2}+ (\sqrt{2})^2 \)
 \( \displaystyle =3- 2\sqrt{6}+ 2=5-2\sqrt{6} \)
 \( \displaystyle \sqrt{27}- \sqrt{6}\sqrt{2}=\sqrt{3^2\times 3}- \sqrt{12} \)
 \( \displaystyle =3\sqrt{3}- \sqrt{2^2\times 3}=3\sqrt{3}-2\sqrt{3}=\sqrt{3} \)
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