Question

Rationalise 1 by root 7 + root 5

Answer 1

We are asked to rationalise,

• ${\underline{\boxed{\displaystyle{\sf{\dfrac{1}{\sqrt{7} + \sqrt{5}}}}}}}$

Now let us rationalise!

$:\implies \sf \dfrac{1}{\sqrt{7} + \sqrt{5}} \\ \\ :\implies \sf \dfrac{1}{\sqrt{7} + \sqrt{5}} \cdot \dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}} \\ \\ \sf Let \: us \: solve \: denominator \: 1^{st} \\ \\ \sf Using \: identity \: \rightarrow (a+b) (a-b) = a^2 - b^2 \\ \\ :\implies \sf (\sqrt{7}^{2}) - (\sqrt{5}^{2}) \\ \\ :\implies \sf 7 - 5 \\ \\ :\implies \sf 2 \\ \\ \sf Now \: let's \: further \: solve \\ \\ :\implies \sf \dfrac{1 \cdot \sqrt{7}-\sqrt{5}}{2} \\ \\ :\implies \sf \dfrac{\sqrt{7}-\sqrt{5}}{2} \\ \\ {\pmb{\sf{Henceforth, \: rationalised!}}}$