8:
c: \(=2\cdot\dfrac{4}{\sqrt{3}}-3\cdot\dfrac{1}{3\sqrt{3}}-6\cdot\dfrac{2}{5\sqrt{3}}\)
\(=\dfrac{8}{\sqrt{3}}-\dfrac{1}{\sqrt{3}}-\dfrac{12}{5\sqrt{3}}\)
\(=\dfrac{7}{\sqrt{3}}-\dfrac{12\sqrt{3}}{15}\)
\(=\dfrac{35\sqrt{3}-12\sqrt{3}}{15}=\dfrac{23\sqrt{3}}{15}\)
d: \(=\sqrt{\dfrac{\left(2-\sqrt{3}\right)^2}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}}+\sqrt{\dfrac{\left(2+\sqrt{3}\right)^2}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}}\)
\(=\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{\left(2+\sqrt{3}\right)^2}\)
\(=2-\sqrt{3}+2+\sqrt{3}\)
=4
9:
a: \(=\dfrac{\sqrt{6-2\sqrt{5}}\left(3+\sqrt{5}\right)}{2\left(\sqrt{5}+1\right)}=\dfrac{\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)}{2\left(\sqrt{5}+1\right)}\)
\(=\dfrac{3\sqrt{5}+5-3-\sqrt{5}}{2\left(\sqrt{5}+1\right)}=\dfrac{2\sqrt{5}+2}{2\left(\sqrt{5}+1\right)}\)
=1
b: \(=\dfrac{4\left(\sqrt{3}+1\right)}{2}+\dfrac{\sqrt{3}+1}{2}-\dfrac{6\left(3+\sqrt{3}\right)}{6}\)
\(=2\left(\sqrt{3}+1\right)+\dfrac{1}{2}\sqrt{3}+\dfrac{1}{2}-3-\sqrt{3}\)
\(=2\sqrt{3}+2-\dfrac{1}{2}\sqrt{3}-\dfrac{5}{2}\)
\(=\dfrac{3}{2}\sqrt{3}-\dfrac{1}{2}\)
c: \(=\sqrt{4-2\sqrt{2}-1}=\sqrt{3-2\sqrt{2}}=\sqrt{2}-1\)
d: \(=\dfrac{\sqrt{2}}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{\sqrt{2}}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\dfrac{\sqrt{2}}{2+\sqrt{3}+1}+\dfrac{\sqrt{2}}{2-\sqrt{3}+1}\)
\(=\dfrac{\sqrt{2}}{3+\sqrt{3}}+\dfrac{\sqrt{2}}{3-\sqrt{3}}=\dfrac{\sqrt{2}\left(3-\sqrt{3}\right)+\sqrt{2}\left(3+\sqrt{3}\right)}{9-3}\)
\(=\dfrac{3\sqrt{2}-\sqrt{6}+3\sqrt{2}+\sqrt{6}}{6}\)
\(=\dfrac{6\sqrt{2}}{6}=\sqrt{2}\)
