Question

Rationalise the denominator.
(i) $\frac{4}{\sqrt{5}}$

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

(iii) $\frac{2}{\sqrt[3]{3}}$

(iv) $\frac{3\sqrt{3}}{2\sqrt{8}}$

(v) $\frac{3\sqrt[3]{5}}{\sqrt[3]{9}}$

(vi) $\frac{5+\sqrt{6}}{5-\sqrt{6}}$

(vii) $\frac{3}{\sqrt{6}-\sqrt{7}}$

(viii) $\frac{2\sqrt{8}}{2\sqrt{5}-\sqrt{3}}$

(ix) $\frac{1}{\sqrt{6}+\sqrt{5}-\sqrt{11}}$

(x) $\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}$.

Answer 1

$\left(\mathrm{i}\right)\frac{4}{\sqrt{5}}=\frac{4}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{4\sqrt{5}}{5}$

$\left(\mathrm{ii}\right)\frac{1}{\sqrt{12}}=\frac{1}{\sqrt{12}}\times \frac{\sqrt{12}}{\sqrt{12}}=\frac{1}{12}\sqrt{12}=\frac{2}{12}\sqrt{3}=\frac{\sqrt{3}}{6}$

$\left(\mathrm{iii}\right)\frac{2}{3\sqrt{3}}=\frac{2}{3\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{9}$

$\left(\mathrm{iv}\right)\frac{3\sqrt{3}}{2\sqrt{8}}=\frac{3\sqrt{3}}{2\sqrt{8}}\times \frac{\sqrt{8}}{\sqrt{8}}=\frac{3\sqrt{24}}{16}=\frac{3\times 2\sqrt{6}}{16}=\frac{3\sqrt{6}}{8}$

$$\begin{gathered} \left(\mathrm{v}\right)\frac{3\sqrt[3]{5}}{\sqrt[3]{9}} \\ =\frac{3\sqrt[3]{5}}{\sqrt[3]{3^2}} \\ =\frac{3\sqrt[3]{5}}{\sqrt[3]{3^2}}\times \frac{\sqrt[3]{3}}{\sqrt[3]{3}} \\ =\frac{3\times \sqrt[3]{5\times 3}}{3} \\ =\sqrt[3]{15} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vi}\right)\frac{5+\sqrt{6}}{5-\sqrt{6}} \\ =\frac{5+\sqrt{6}}{5-\sqrt{6}}\times \frac{5+\sqrt{6}}{5+\sqrt{6}} \\ =\frac{{\left(5+\sqrt{6}\right)}^2}{{\left(5\right)}^2-{\left(\sqrt{6}\right)}^2} \\ =\frac{25+6+10\sqrt{6}}{25-6} \\ =\frac{31+10\sqrt{6}}{19} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vii}\right)\frac{3}{\sqrt{6}-\sqrt{7}} \\ =\frac{3}{\sqrt{6}-\sqrt{7}}\times \frac{\sqrt{6}+\sqrt{7}}{\sqrt{6}+\sqrt{7}} \\ =\frac{3\sqrt{6}+3\sqrt{7}}{{\left(\sqrt{6}\right)}^2-{\left(\sqrt{7}\right)}^2} \\ =\frac{3\sqrt{6}+3\sqrt{7}}{-1} \\ =-3\left(\sqrt{6}+\sqrt{7}\right) \end{gathered}$$

$$\begin{gathered} \left(\mathrm{viii}\right)\frac{2\sqrt{8}}{2\sqrt{5}-\sqrt{3}} \\ =\frac{2\sqrt{4}\times \sqrt{2}}{2\sqrt{5}-\sqrt{3}}\times \frac{2\sqrt{5}+\sqrt{3}}{2\sqrt{5}+\sqrt{3}} \\ =\frac{4\sqrt{2}\left(2\sqrt{5}+\sqrt{3}\right)}{{\left(2\sqrt{5}\right)}^2-{\left(\sqrt{3}\right)}^2} \\ =\frac{8\sqrt{10}+4\sqrt{6}}{20-3} \\ =\frac{8\sqrt{10}+4\sqrt{6}}{17} \\ =\frac{4\times 2\times \sqrt{2}\times \sqrt{5}+4\sqrt{2}\times \sqrt{3}}{17} \\ =\frac{4\sqrt{2}\left(2\sqrt{5}+\sqrt{3}\right)}{17} \end{gathered}$$


$$\begin{gathered} \left(\mathrm{ix}\right)\frac{1}{\sqrt{6}+\sqrt{5}-\sqrt{11}} \\ =\frac{1}{\left(\sqrt{6}+\sqrt{5}\right)-\sqrt{11}}\times \frac{\left(\sqrt{6}+\sqrt{5}\right)+\sqrt{11}}{\left(\sqrt{6}+\sqrt{5}\right)+\sqrt{11}} \\ =\frac{\sqrt{6}+\sqrt{5}+\sqrt{11}}{{\left(\sqrt{6}+\sqrt{5}\right)}^2-{\left(\sqrt{11}\right)}^2} \\ =\frac{\sqrt{6}+\sqrt{5}-\sqrt{11}}{6+5+2\sqrt{30}-11} \\ =\frac{\sqrt{6}+\sqrt{5}-\sqrt{11}}{2\sqrt{30}}\times \frac{\sqrt{30}}{\sqrt{30}} \\ =\frac{\sqrt{30}\left(\sqrt{6}+\sqrt{5}-\sqrt{11}\right)}{60} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{x}\right)\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}} \\ =\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}\times \frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}} \\ =\frac{{\left(\sqrt{a+x}+\sqrt{a-x}\right)}^2}{{\left(\sqrt{a+x}\right)}^2-{\left(\sqrt{a-x}\right)}^2} \\ =\frac{a+x+a-x+2\sqrt{\left(a+x\right)\left(a-x\right)}}{a+x-a+x} \\ =\frac{2a+2\sqrt{a^2-x^2}}{2x} \\ =\frac{2\left(a+\sqrt{a^2-x^2}\right)}{2x} \\ =\frac{a+\sqrt{a^2-x^2}}{x} \end{gathered}$$