a) \(\sqrt{\dfrac{1}{8}}\cdot\sqrt{2}\cdot\sqrt{125}\cdot\sqrt{\dfrac{1}{5}}\)
\(=\dfrac{1}{\sqrt{8}}\cdot\sqrt{2}\cdot5\sqrt{5}\cdot\dfrac{1}{\sqrt{5}}\)
\(=\dfrac{\sqrt{2}}{2\sqrt{2}}\cdot\dfrac{5\sqrt{5}}{\sqrt{5}}\)
\(=\dfrac{1}{2}\cdot5\)
\(=\dfrac{5}{2}\)
b) \(4\sqrt{50}+2\sqrt{8}-4\sqrt{72}-\sqrt{32}\)
\(=4\cdot5\sqrt{2}+2\cdot2\sqrt{2}-4\cdot6\sqrt{2}-4\sqrt{2}\)
\(=20\sqrt{2}+4\sqrt{2}-24\sqrt{2}-4\sqrt{2}\)
\(=\left(20+4-24-4\right)\sqrt{2}\)
\(=-4\sqrt{2}\)
c) \(2\sqrt{20}-3\sqrt{45}+5\sqrt{80}-5\sqrt{5}\)
\(=2\cdot2\sqrt{5}-3\cdot3\sqrt{5}+5\cdot4\sqrt{5}-5\sqrt{5}\)
\(=4\sqrt{5}-9\sqrt{5}+20\sqrt{5}-5\sqrt{5}\)
\(=\left(20-9-5+4\right)\sqrt{5}\)
\(=10\sqrt{5}\)
d) \(2ab\sqrt{a^2b}-5a^2\sqrt{b^3}\) (\(a,b\ge0\))
\(=2ab\cdot\left|a\right|\sqrt{b}-5a^2\left|b\right|\sqrt{b}\)
\(=2a^2b\sqrt{b}-5a^2b\sqrt{b}\)
\(=\left(2a^2b-5a^2b\right)\sqrt{b}\)
\(=-3a^2b\sqrt{b}\)
e) \(\sqrt{40}+\sqrt{\dfrac{2}{5}}-\sqrt{\dfrac{5}{2}}\)
\(=2\sqrt{10}+\dfrac{\sqrt{10}}{5}-\dfrac{\sqrt{10}}{2}\)
\(=\dfrac{20\sqrt{10}}{10}+\dfrac{2\sqrt{10}}{10}-\dfrac{5\sqrt{10}}{10}\)
\(=\dfrac{\left(20+2-5\right)\sqrt{10}}{10}\)
\(=\dfrac{17\sqrt{10}}{10}\)
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