Question

Simplify by rationalizing the denominator.
(i) $\frac{1}{11+\sqrt{3}}$
(ii) $\frac{1}{9+3\sqrt{5}}$
(iii) $\frac{1}{\sqrt{11}+\sqrt{13}}$
(iv)$\frac{\sqrt{5}+1}{\sqrt{5}-1}$
(v) $\frac{3-\sqrt{3}}{2+5\sqrt{3}}$.

Answer 1

$\left(i\right)\frac{1}{11+\sqrt{3}}\times \frac{11-\sqrt{3}}{11-\sqrt{3}}=\frac{11-\sqrt{3}}{118}$

$\left(ii\right)\frac{1}{9+3\sqrt{5}}\times \frac{9-3\sqrt{5}}{9-3\sqrt{5}}=\frac{9-3\sqrt{5}}{36}$

$\left(iii\right)\frac{1}{\sqrt{11}+\sqrt{13}}\times \frac{\sqrt{11}-\sqrt{13}}{\sqrt{11}-\sqrt{13}}=\frac{\sqrt{11}-\sqrt{13}}{-2}$

$\left(iv\right)\frac{\sqrt{5}+1}{\sqrt{5}-1}\times \frac{\sqrt{5}+1}{\sqrt{5}+1}=\frac{6+2\sqrt{5}}{4}$

$\left(v\right)\frac{3-\sqrt{3}}{2+5\sqrt{3}}\times \frac{2-5\sqrt{3}}{2-5\sqrt{3}}=\frac{(3-\sqrt{3})(2-5\sqrt{3})}{-71}$