Question

Rationalize the denominators of $\frac{1}{\sqrt{7}-\sqrt{6}}$

Answer 1

Given expression

$\frac{1}{\sqrt{7}-\sqrt{6}}$

To rationalize it we have to multiply the denominator and numerator by $\frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}$

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

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

Hence we get $\sqrt{7}+\sqrt{6}$ after rationalization.