a) \(\sqrt[]{48}+\sqrt[]{5\dfrac{1}{3}}+2\sqrt[]{75}-5\sqrt[]{1\dfrac{1}{3}}\)
\(=4\sqrt[]{3}+\sqrt[]{\dfrac{16}{3}}+2.5\sqrt[]{3}-5\sqrt[]{\dfrac{4}{3}}\)
\(=4\sqrt[]{3}+10\sqrt[]{3}+\dfrac{4}{\sqrt[]{3}}+\dfrac{10}{\sqrt[]{3}}\)
\(=14\sqrt[]{3}+\dfrac{14}{\sqrt[]{3}}\)
\(=14\left(\sqrt[]{3}+\dfrac{1}{\sqrt[]{3}}\right)\)
b) \(\sqrt[]{\left(2-\sqrt[]{3}\right)^2}+\sqrt[]{4-2\sqrt[]{3}}\)
\(=\left|2-\sqrt[]{3}\right|+\sqrt[]{3-2\sqrt[]{3}+1}\)
\(=\left|2-\sqrt[]{3}\right|+\sqrt[]{\left(\sqrt[]{3}-1\right)^2}\)
\(=\left|2-\sqrt[]{3}\right|+\left|\sqrt[]{3}-1\right|\)
\(=2-\sqrt[]{3}+\sqrt[]{3}-1\left(2>\sqrt[]{3};\sqrt[]{3}>1\right)\)
\(=1\)
c) \(\left(\sqrt[]{10}-\sqrt[]{2}\right)\sqrt[]{3+\sqrt[]{5}}\)
\(=\sqrt[]{2}\left(\sqrt[]{5}-1\right)\sqrt[]{3+\sqrt[]{5}}\)
\(=\left(\sqrt[]{5}-1\right)\sqrt[]{6+2\sqrt[]{5}}\)
\(=\left(\sqrt[]{5}-1\right)\sqrt[]{5+2\sqrt[]{5}+1}\)
\(=\left(\sqrt[]{5}-1\right)\sqrt[]{\left(\sqrt[]{5}+1\right)^2}\)
\(=\left(\sqrt[]{5}-1\right)\left|\sqrt[]{5}+1\right|\)
\(=\left(\sqrt[]{5}-1\right)\left(\sqrt[]{5}+1\right)=5-1=4\)
e) \(3\sqrt[]{\left(a-2\right)^2}-\sqrt[]{9a^2}+\sqrt[]{\dfrac{4a^2}{25}}\left(a< 0\right)\)
\(=3\left|a-2\right|-\sqrt[]{\left(3a\right)^2}+\sqrt[]{\left(\dfrac{2a}{5}\right)^2}\)
\(=3\left|a-2\right|-\left|3a\right|+\left|\dfrac{2a}{5}\right|\)
\(=3\left(2-a\right)+3a-\dfrac{2a}{5}\left(a< 0\right)\)
\(=6-3a+3a-\dfrac{2a}{5}\)
\(=6-\dfrac{2a}{5}=2\left(3-\dfrac{a}{5}\right)\)
