Rút gọn các biểu thức
a) \(\sqrt{\left(2-\sqrt{3}\right)}^2\) + \(\sqrt{4-2\sqrt{3}}\)
b) \(\sqrt{15-6\sqrt{6}}\) + \(\sqrt{33-12\sqrt{6}}\)
c) \(\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}\) + \(\sqrt{\frac{3+\sqrt{5}}{3-\sqrt{5}}}\)
d)\(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}\) + \(\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)
\(a)\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{4-2\sqrt{3}}\)
\(=2-\sqrt{3}+\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=2-\sqrt{3}+\sqrt{3}-1=1\)
\(b)\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{33-2.3.\sqrt{4}.\sqrt{6}}\)
\(=3-\sqrt{6}+\sqrt{33-2.3.\sqrt{24}}\)
\(=3-\sqrt{6}+\sqrt{\left(\sqrt{24}-3\right)^2}\)
\(=3-\sqrt{6}+\sqrt{24}-3\)
\(=\sqrt{24}-\sqrt{6}\)
\(=\sqrt{6}\left(2-1\right)=\sqrt{6}\)
\(c)\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}+\sqrt{\frac{3+\sqrt{5}}{3-\sqrt{5}}}\)
\(=\sqrt{\frac{\left(3-\sqrt{5}\right)^2}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}}+\sqrt{\frac{\left(3+\sqrt{5}\right)^2}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}}\)
\(=\sqrt{\frac{\left(3-\sqrt{5}\right)^2}{4}}+\sqrt{\frac{\left(3+\sqrt{5}\right)^2}{4}}\)
\(=\frac{3-\sqrt{5}}{2}+\frac{3+\sqrt{5}}{2}\)
\(=\frac{6}{2}=3\)
\(d)\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)
\(=\frac{\left(\sqrt{7}+\sqrt{5}\right)^2+\left(\sqrt{7}-\sqrt{5}\right)^2}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}\)
\(=\frac{24}{2}=12\)
