[tex]1. \frac{8}{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } = \frac{8 \sqrt{2} }{ {2} } = 4 \sqrt{2} [/tex]
[tex] 2.\sqrt[3]{ \frac{5}{4} } \times \sqrt[3]{ \frac{4^{2} }{ {4}^{2} } } = \sqrt[3]{ \frac{80}{64} } = \frac{2 \sqrt{10} }{4} \: or \frac{ \sqrt{10} }{2} [/tex]
[tex]3. \sqrt[4]{ \frac{5}{8} } = \sqrt[4]{ \frac{5}{ {2}^{3} } } \times \sqrt[4]{ \frac{2}{2} } = \sqrt[4]{ \frac{10}{16} } = \frac{ \sqrt[4]{10} }{2} [/tex]
[tex]4. \frac{14}{ \sqrt{7}} \times \sqrt{ \frac{7}{7} } = \frac{14 \sqrt{7} }{ \sqrt{49} } = \frac{14 \sqrt{7} }{7} or \: 2 \sqrt{7} [/tex]
[tex]5. \frac{ \sqrt[3]{9} }{ \sqrt[3]{6} } \times \sqrt[3]{ \frac{6 {}^{2} }{ {6}^{2} } } = \frac{ \sqrt[3]{324} }{ \sqrt[3]{216} } = \frac{3 \sqrt{12} }{6} \: or \: \frac{ \sqrt[3]{12} }{2} [/tex]
[tex]6. \frac{1}{5 - \sqrt{2} } \times \frac{5 + \sqrt{2} }{5 + \sqrt{2} } = \frac{5 + \sqrt{2} }{25 - 2} = \frac{5 + \sqrt{2} }{23} [/tex]
[tex]7. \frac{4}{ \sqrt{6} + 2} \times \frac{ \sqrt{6} - 2}{ \sqrt{6} - 2} = \frac{4( \sqrt{6} - 2) }{6 - 4} = 2 \sqrt{6} - 4[/tex]
[tex]8. \frac{2}{ \sqrt{7} - \sqrt{3} } \times \frac{( \sqrt{7} + \sqrt{3}) }{ \sqrt{7} + \sqrt{3} } = \frac{2( \sqrt{7} + \sqrt{3}) }{7 - 3} = \frac{ \sqrt{7} + \sqrt{3} }{2} [/tex]
[tex]9. \frac{5}{ - 4 + \sqrt{5} } = \frac{5}{ \sqrt{5} - 4} \times \frac{ \sqrt{5} + 4 }{ \sqrt{5} + 4 } = \frac{5( \sqrt{5} + 4) }{5 - 16} = - \frac{5 \sqrt{5} + 20}{11} [/tex]
[tex]10. \frac{ \sqrt{2} }{ \sqrt{2} - 1 } \times \frac{ \sqrt{2} + 1 }{ \sqrt{2} + 1} = \frac{2 + \sqrt{2} }{2 - 1} = 2 + \sqrt{2} [/tex]
Hope it helps!
