a) \(\dfrac{5\sqrt{2}+\sqrt{10}}{\sqrt{5}+1}+\dfrac{12}{4-\sqrt{10}}-8\sqrt{\dfrac{5}{2}}\)
\(=\dfrac{\sqrt{10}\left(\sqrt{5}+1\right)}{\sqrt{5}+1}+\dfrac{12\left(4+\sqrt{10}\right)}{6}-\dfrac{8\sqrt{10}}{2}\)
\(=\sqrt{10}+2\left(4+\sqrt{10}\right)-4\sqrt{10}\\ =8-\sqrt{10}\)
b) \(\dfrac{\sqrt{3}+4}{5-2\sqrt{3}}-\dfrac{5\sqrt{3}-3}{\sqrt{3}}+6\sqrt{\dfrac{1}{3}}\)
\(=\dfrac{\left(\sqrt{3}+4\right)\left(5+2\sqrt{3}\right)}{13}-\left(5-\sqrt{3}\right)+\dfrac{6\sqrt{3}}{3}\)
\(=\dfrac{13\sqrt{3}+26}{13}-5+\sqrt{3}+2\sqrt{3}\\ =2+\sqrt{3}-5+\sqrt{3}+2\sqrt{3}\\ =4\sqrt{3}-3\)
`(a):(5\sqrt{2}+\sqrt{10})/(\sqrt{5}+1)+(12)/(4-\sqrt{10})-8\sqrt{(5)/(2)}`
`=(\sqrt{25.2}+\sqrt{10})/(\sqrt{5}+1)+(12(4+\sqrt{10}))/((4+\sqrt{10})(4-\sqrt{10}))-\sqrt{(64.5)/(2)}`
`=(\sqrt{10}.(\sqrt{5}+1))/(\sqrt{5}+1)+(12(4+\sqrt{10}))/(16-10)-\sqrt{160}`
`=\sqrt{10}+2(4+\sqrt{10})-4\sqrt{10}`
`=\sqrt{10}+8+2\sqrt{10}-4\sqrt{10}`
`=8-\sqrt{10}`
`(b):(\sqrt{3}+4)/(5-2\sqrt{3})-(5\sqrt{3}-3)/(\sqrt{3})+6\sqrt{1/3}`
`=((\sqrt{3}+4)(5+2\sqrt{3}))/((5-2\sqrt{3})(5+2\sqrt{3}))-(\sqrt{3}(5-\sqrt{3}))/(\sqrt{3})+\sqrt{(36)/(3)}`
`=(26+13\sqrt{3})/(25-12)-(5-\sqrt{3})+\sqrt{12}`
`=(13(2+\sqrt{3}))/(13)-5+\sqrt{3}+2\sqrt{3}`
`=2+\sqrt{3}-5+\sqrt{3}+2\sqrt{3}`
`=4\sqrt{3}-3`
`a)[5\sqrt{2}+\sqrt{10}]/[\sqrt{5}+1]+12/[4-\sqrt{10}]-8\sqrt{5/2}`
`=[\sqrt{10}(\sqrt{5}+1)]/[\sqrt{5}+1]+[12(4+\sqrt{10})]/[16-10]-4\sqrt{10}`
`=\sqrt{10}+[12(4+\sqrt{10})]/6-4\sqrt{10}`
`=\sqrt{10}+2(4+s\qrt{10})-4\sqrt{10}`
`=\sqrt{10}+8+2\sqrt{10}-4\sqrt{10}=8-\sqrt{10}`
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`b)[\sqrt{3}+4]/[5-2\sqrt{3}]-[5\sqrt{3}-3]/\sqrt{3}+6\sqrt{1/3}`
`=[(\sqrt{3}+4)(5+2\sqrt{3})]/[25-12]-[\sqrt{3}(5-\sqrt{3})]/\sqrt{3}+2\sqrt{3}`
`=[5\sqrt{3}+6+20+8\sqrt{3}]/13-5+\sqrt{3}+2\sqrt{3}`
`=[13\sqrt{3}+26]/13-5+3\sqrt{3}`
`=\sqrt{3}+2-5+3\sqrt{3}=4\sqrt{3}-3`
