Selina Solutions Class 9 Mathematics Solutions for Exercise 1(C) in Chapter 1 - Chapter 1- Rational and Irrational Numbers

Question 11 Exercise 1(C)

Show that:

$\frac{1}{3-2 \sqrt{2}}-\frac{1}{2 \sqrt{2}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}=5$

Answer:

$\begin{aligned} \text { L.H.S. } &=\frac{1}{3-2 \sqrt{2}}-\frac{1}{2 \sqrt{2}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2} \\ &=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2} \\ &=\frac{1}{3-\sqrt{8}} \times \frac{3+\sqrt{8}}{3+\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}} \times \frac{\sqrt{8}+\sqrt{7}}{\sqrt{8}+\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}} \times \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}} \\ &=\frac{1}{\sqrt{6}-\sqrt{5}} \times \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}}+\frac{1}{\sqrt{5}-2} \times \frac{\sqrt{5}+2}{\sqrt{5}+2} \\ &=\frac{3+\sqrt{8}}{(3)^{2}-(\sqrt{8})^{2}}-\frac{\sqrt{8}+\sqrt{7}}{(\sqrt{8})^{2}-(\sqrt{7})^{2}}+\frac{\sqrt{7}+\sqrt{6}}{(\sqrt{7})^{2}-(\sqrt{6})^{2}}-\frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}+\frac{\sqrt{5}+2}{(\sqrt{5})^{2}-(2)^{2}} \\ &=3+\sqrt{8}-\sqrt{8}-\sqrt{7}+\sqrt{7}+\sqrt{6}-\sqrt{6}-\sqrt{5}+\sqrt{5}+2 \\ &=3+2 \\ &=5 \\ &=\mathrm{R.H.S.} \end{aligned}$