Rationalization on surds | Denominator contains a irrational numbers

Rationalization of Denominators

Illustrations: Rationalise the denomination in each of the following case: i) \(\frac{1}{\sqrt{5}}\) ii) \(\frac{1}{5+\sqrt{2}}\) iii) \(\frac{2}{\sqrt{5}-\sqrt{3}}\) iv) \(\frac{1}{9-4\sqrt{5}}\)

Solve:

i) \( \frac{1}{\sqrt{5}} \)

Multiply numerator and denominator by \( \sqrt{5} \):
\( \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \)
Answer: \( \boxed{\frac{\sqrt{5}}{5}} \)

ii) \( \frac{1}{5 + \sqrt{2}} \)

Multiply numerator and denominator by the conjugate \( 5 - \sqrt{2} \):
\( \frac{1}{5+\sqrt{2}} \times \frac{5 - \sqrt{2}}{5 - \sqrt{2}} = \frac{5 - \sqrt{2}}{(5 + \sqrt{2})(5 - \sqrt{2})} \)
\( = \frac{5 - \sqrt{2}}{25 - 2} = \frac{5 - \sqrt{2}}{23} \)
Answer: \( \boxed{\frac{5 - \sqrt{2}}{23}} \)

iii) \( \frac{2}{\sqrt{5} - \sqrt{3}} \)

Multiply numerator and denominator by the conjugate \( \sqrt{5} + \sqrt{3} \):
\( \frac{2}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{2(\sqrt{5} + \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2} \)
\( = \frac{2(\sqrt{5} + \sqrt{3})}{5 - 3} = \frac{2(\sqrt{5} + \sqrt{3})}{2} = \sqrt{5} + \sqrt{3} \)
Answer: \( \boxed{\sqrt{5} + \sqrt{3}} \)

iv) \( \frac{1}{9 - 4\sqrt{5}} \)

Multiply numerator and denominator by the conjugate \( 9 + 4\sqrt{5} \):
\( \frac{1}{9 - 4\sqrt{5}} \times \frac{9 + 4\sqrt{5}}{9 + 4\sqrt{5}} = \frac{9 + 4\sqrt{5}}{(9)^2 - (4\sqrt{5})^2} \)
\( = \frac{9 + 4\sqrt{5}}{81 - 80} = \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5} \)
Answer: \( \boxed{9 + 4\sqrt{5}} \)