[tex]a) \frac{5}{ \sqrt{3} } = \frac{5 \sqrt{3} }{ \sqrt{3} \sqrt{3} } = \frac{5 \sqrt{3} }{3} [/tex]
[tex]b) \frac{10}{ \sqrt{5} } = \frac{10 \sqrt{5} }{ \sqrt{5} \sqrt{5} } = \frac{10 \sqrt{5} }{5} = 2 \sqrt{5} [/tex]
[tex]c) \frac{ - 6}{ \sqrt{6} } = \frac{ - 6 \sqrt{6} }{ \sqrt{6} \sqrt{6} } = \frac{ - 6 \sqrt{6} }{6} = - \sqrt{6} [/tex]
[tex]d) \frac{6 \sqrt{3} }{ - \sqrt{2} } = - \frac{6 \sqrt{3} }{ \sqrt{2} } = - \frac{6 \sqrt{3} \sqrt{2} }{ \sqrt{2} \sqrt{2} } = - \frac{6 \sqrt{6} }{2} = - 3 \sqrt{6} [/tex]
[tex]e) \frac{8 \sqrt{3} }{2 \sqrt{6} } = \frac{4 \sqrt{3} }{ \sqrt{3} \sqrt{2} } = \frac{4}{ \sqrt{2} } = \frac{4 \sqrt{2} }{ \sqrt{2} \sqrt{2} } = \frac{4 \sqrt{2} }{2} = 2 \sqrt{2} [/tex]
[tex]f) \frac{4 \sqrt{3} + \sqrt{12} }{ \sqrt{3} } = \frac{4 \sqrt{3} + 2 \sqrt{3} }{ \sqrt{3} } = \frac{6 \sqrt{3} }{ \sqrt{3} } = 6 [/tex]
[tex]g) \frac{15 \sqrt{3} + 3 }{ \sqrt{75} } = \frac{15 \sqrt{3} + 3 }{5 \sqrt{3} } = \frac{(15 \sqrt{3} + 3) \sqrt{3} }{5 \sqrt{3} \sqrt{3} } = \frac{3(5 \sqrt{3} + 1) \sqrt{3} }{5 \times 3} = \frac{3(5 \sqrt{3} + 1) }{15} = \frac{(5 \sqrt{3} + 1) \sqrt{3} }{5} = \frac{15 + \sqrt{3} }{5} [/tex]
[tex]h) \frac{1}{2 \sqrt{3} + 5 \sqrt{3} } = \frac{1}{7 \sqrt{3} } = \frac{1 \sqrt{3} }{7 \sqrt{3} \sqrt{3} } = \frac{ \sqrt{3} }{7 \times 3} = \frac{ \sqrt{3} }{21} [/tex]
[tex]i) \frac{3x}{ \sqrt{3x} },x > 0 = \frac{3x \sqrt{3x} }{ \sqrt{3x} \sqrt{3x} } = \frac{3x \sqrt{3x} }{3x} = \frac{x \sqrt{3x} }{x} = \sqrt{3x} [/tex]
