Answer:
1) [tex]\dfrac{2}{\sqrt{5} } = \dfrac{2 \cdot \sqrt{5} }{5}[/tex]
2) [tex]-\dfrac{5}{\sqrt{3} } = -\dfrac{5 \cdot \sqrt{3} }{3}[/tex]
3) [tex]\dfrac{\sqrt{2} + \sqrt{5} }{\sqrt{10} } =\dfrac{\sqrt{5} }{5} + \dfrac{ \sqrt{2} }{2}[/tex]
4) [tex]\dfrac{3 + \sqrt{2} }{\sqrt{3} } \times \dfrac{\sqrt{3} }{\sqrt{3} } = \sqrt{3} + \dfrac{\sqrt{6} }{3}[/tex]
5) [tex]\dfrac{\sqrt{3} }{\sqrt{5} + \sqrt{2} }= \dfrac{\sqrt{15} - \sqrt{6} }{3}[/tex]
Step-by-step explanation:
The rationalization of the denominator of the surds are found as follows;
1) [tex]\dfrac{2}{\sqrt{5} }[/tex]
[tex]\dfrac{2}{\sqrt{5} } \times \dfrac{\sqrt{5} }{\sqrt{5} } = \dfrac{2 \cdot \sqrt{5} }{5}[/tex]
[tex]\dfrac{2}{\sqrt{5} } = \dfrac{2 \cdot \sqrt{5} }{5}[/tex]
2) [tex]-\dfrac{5}{\sqrt{3} }[/tex]
[tex]-\dfrac{5}{\sqrt{3} } \times \dfrac{\sqrt{3} }{\sqrt{3} } = -\dfrac{5 \cdot \sqrt{3} }{3}[/tex]
[tex]-\dfrac{5}{\sqrt{3} } = -\dfrac{5 \cdot \sqrt{3} }{3}[/tex]
3) [tex]\dfrac{\sqrt{2} + \sqrt{5} }{\sqrt{10} }[/tex]
[tex]\dfrac{\sqrt{2} + \sqrt{5} }{\sqrt{10} } \times \dfrac{ \sqrt{10} }{\sqrt{10} } = \dfrac{\sqrt{20} + \sqrt{50} }{10 } = \dfrac{2\cdot \sqrt{5} + 5 \cdot \sqrt{2} }{10} = \dfrac{\sqrt{5} }{5} + \dfrac{ \sqrt{2} }{2}[/tex]
[tex]\dfrac{\sqrt{2} + \sqrt{5} }{\sqrt{10} } =\dfrac{\sqrt{5} }{5} + \dfrac{ \sqrt{2} }{2}[/tex]
4) [tex]\dfrac{3 + \sqrt{2} }{\sqrt{3} }[/tex]
[tex]\dfrac{3 + \sqrt{2} }{\sqrt{3} } \times \dfrac{\sqrt{3} }{\sqrt{3} } = \dfrac{3 \cdot \sqrt{3}+\sqrt{6} }{3 } = \sqrt{3} + \dfrac{\sqrt{6} }{3}[/tex]
[tex]\dfrac{3 + \sqrt{2} }{\sqrt{3} } \times \dfrac{\sqrt{3} }{\sqrt{3} } = \sqrt{3} + \dfrac{\sqrt{6} }{3}[/tex]
5) [tex]\dfrac{\sqrt{3} }{\sqrt{5} + \sqrt{2} }[/tex]
[tex]\dfrac{\sqrt{3} }{\sqrt{5} + \sqrt{2} } = \dfrac{\sqrt{5} - \sqrt{2} }{\sqrt{5} - \sqrt{2} } = \dfrac{\sqrt{15} -\sqrt{6} }{5 - 2} = \dfrac{\sqrt{15} - \sqrt{6} }{3}[/tex]
[tex]\dfrac{\sqrt{3} }{\sqrt{5} + \sqrt{2} }= \dfrac{\sqrt{15} - \sqrt{6} }{3}[/tex]
6) [tex]\dfrac{\sqrt{7} }{\sqrt{3} - \sqrt{5} }[/tex]
[tex]\dfrac{\sqrt{7} }{\sqrt{3} - \sqrt{5} } \times \dfrac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \dfrac{\sqrt{21} + \sqrt{35}}{{3} + {5}} = \dfrac{\sqrt{21} + \sqrt{35}}{8}[/tex]
[tex]\dfrac{\sqrt{7} }{\sqrt{3} - \sqrt{5} } \times \dfrac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} =\dfrac{\sqrt{21} + \sqrt{35}}{8}[/tex]
