# Rationalize the denominator of each of the following:

Question:

Rationalize the denominator of each of the following:

(i) $\frac{3}{\sqrt{5}}$

(ii) $\frac{3}{2 \sqrt{5}}$

(iii) $\frac{1}{\sqrt{12}}$

(iv) $\frac{\sqrt{2}}{\sqrt{3}}$

(v) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}$

(vi) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}$

(vii) $\frac{3 \sqrt{2}}{5}$

Solution:

(i) $\frac{3}{\sqrt{5}}$

For rationalizing the denominator, multiply both numerator and denominator with $\sqrt{5}$

$=\frac{3 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}=\frac{3 \times \sqrt{5}}{\sqrt{5}}$

(ii) $\frac{3}{2 \sqrt{5}}$

For rationalizing the denominator, multiply both numerator and denominator with √5

$=\frac{3 \times \sqrt{5}}{2 \sqrt{5} \times \sqrt{5}}$

$=\frac{3 \sqrt{5}}{2 \times \sqrt{5 \times 5}}$

$=\frac{3 \sqrt{5}}{2 \times 5}$

$=\frac{3 \sqrt{5}}{10}$

(iii) $\frac{1}{\sqrt{12}}$

For rationalizing the denominator, multiply both numerator and denominator with $\sqrt{1} 2$

$=\frac{1 \times \sqrt{12}}{\sqrt{12} \times \sqrt{12}}$

$=\frac{\sqrt{12}}{\sqrt{12 \times 12}}$

$=\frac{\sqrt{12}}{12}$

(iv) $\frac{\sqrt{2}}{\sqrt{3}}$

For rationalizing the denominator, multiply both numerator and denominator with $\sqrt{3}$

$=\frac{\sqrt{2} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

$=\frac{\sqrt{2 \times 3}}{\sqrt{3 \times 3}}$

$=\frac{\sqrt{6}}{3}$

(v) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}$

For rationalizing the denominator, multiply both numerator and denominator with $\sqrt{2}$

$=\frac{(\sqrt{3}+1) \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$

$=\frac{(\sqrt{3} \times \sqrt{2})+\sqrt{2}}{\sqrt{2 \times 2}}$

$=\frac{\sqrt{6}+\sqrt{2}}{2}$

(vi) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}$

For rationalizing the denominator, multiply both numerator and denominator with

$=\frac{(\sqrt{2}+\sqrt{5}) \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

$=\frac{(\sqrt{2} \times \sqrt{3})+(\sqrt{5} \times \sqrt{3})}{\sqrt{3} \times \sqrt{3}}$

$=\frac{\sqrt{6}+\sqrt{15}}{3}$

(vii) $\frac{3 \sqrt{2}}{5}$

For rationalizing the denominator, multiply both numerator and denominator with $\sqrt{5}$

$=\frac{3 \sqrt{2} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}$

$=\frac{3 \sqrt{2 \times 5}}{\sqrt{5 \times 5}}$

$=\frac{3 \sqrt{10}}{5}$