次の公式がよく使われます．
○a,b>0のとき\( \displaystyle \sqrt{a}\sqrt{b}=\sqrt{ab} \)
○a,k>0のとき\( \displaystyle \sqrt{a^2\times k}=a\sqrt{k} \)

\( \displaystyle (\sqrt{8}+ 2)(\sqrt{2}-1)=\sqrt{8}\sqrt{2}-\sqrt{8}+2\sqrt{2}-\sqrt{2} \)
\( \displaystyle =\sqrt{16}-\sqrt{8}+ 2\sqrt{2}-2=4-2\sqrt{2}+2\sqrt{2}-2=2 \) \( \displaystyle \sqrt{3}(\sqrt{6}+ 2)-\sqrt{12}=\sqrt{18}+ 2\sqrt{3}-\sqrt{12} \)
\( \displaystyle =3\sqrt{2}+ 2\sqrt{3}-2\sqrt{3}=3\sqrt{2} \) \( \displaystyle (\sqrt{3}- \sqrt{2})(\sqrt{6}+ 2)=\sqrt{18}+ 3\sqrt{3}-\sqrt{12}-2\sqrt{2} \)
\( \displaystyle =3\sqrt{2}+ 2\sqrt{3}-2\sqrt{3}-2\sqrt{2}=\sqrt{2} \)
次のように変形することもできる
\( \displaystyle \sqrt{2}(\sqrt{3}- \sqrt{2})(\sqrt{3}+ \sqrt{2})=\sqrt{2}\times 1=\sqrt{2} \) \( \displaystyle (2-\sqrt{5})(3+ \sqrt{5})=6+ 2\sqrt{5}-3\sqrt{5}-5 \)
\( \displaystyle =1-\sqrt{5} \) \( \displaystyle \sqrt{2}(\sqrt{18}- \sqrt{2})=\sqrt{36}- 2 \)
\( \displaystyle =6-2=4 \) \( \displaystyle \sqrt{3}(\sqrt{6}+ 1)-3\sqrt{2}=\sqrt{18}+ \sqrt{3}- 3\sqrt{2} \)
\( \displaystyle =3\sqrt{2}+ \sqrt{3}-3\sqrt{2}=\sqrt{3} \) \( \displaystyle (3- \sqrt{7})^2=9-6\sqrt{7}+ 7 \)
\( \displaystyle =16-6\sqrt{7} \) \( \displaystyle \sqrt{2}(\sqrt{24}-\sqrt{6})=\sqrt{48}-\sqrt{12} \)
\( \displaystyle =4\sqrt{3}-2\sqrt{3}=2\sqrt{3} \) \( \displaystyle (\sqrt{5}- \sqrt{2})^2+ \frac{20}{\sqrt{10}}=5-2\sqrt{10}+ 2+ \frac{20\sqrt{10}}{10} \)
\( \displaystyle =7-2\sqrt{10}+ 2\sqrt{10}=7 \) \( \displaystyle (2\sqrt{3}- 1)^2=12-4\sqrt{3}+ 1 \)
\( \displaystyle =13-4\sqrt{3} \) →解説を隠す←