Question

Rationalize the denominators of $\frac{1}{\sqrt{7}-2}$

Answer 1

Rationalization of irrational number:

Given expression

$\frac{1}{\sqrt{7}-2}$

To rationalize it we have to multiply the denominator and numerator by $\frac{\sqrt{7}+2}{\sqrt{7}+2}$

We have to use identity here $\mathbf{(}\mathbf{a}\mathbf{}\mathbf{+}\mathbf{}\mathbf{b}\mathbf{)}\mathbf{(}\mathbf{a}\mathbf{}\mathbf{-}\mathbf{}\mathbf{b}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{a}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathbf{b}}^{\mathbf{2}}$

$$\begin{gathered} \Rightarrow \frac{1}{(\sqrt{7}-2)}\times \frac{\sqrt{7}+2}{(\sqrt{7}+2)}=\frac{\sqrt{7}+2}{{\left(\sqrt{7}\right)}^2-2^2} \\ =\frac{\sqrt{7}+2}{7-4}=\frac{\sqrt{7}+2}{3} \end{gathered}$$

Hence we get $\frac{\sqrt{7}+2}{3}$ after rationalization.