Question

$\frac{1}{\left(\sqrt{9}-\sqrt{8}\right)}-\frac{1}{\left(\sqrt{8}-\sqrt{7}\right)}+\frac{1}{\left(\sqrt{7}-\sqrt{6}\right)}-\frac{1}{\left(\sqrt{6}-\sqrt{5}\right)}+\frac{1}{\left(\sqrt{5}-\sqrt{4}\right)}$ is equal to:

• A. 0
• B. 1/3
• C. 1
• D. 5
• E. 4

Answer 1

The correct option is D

5

$\begin{array}{cccc}=\frac{1}{\left(\sqrt{9}-\sqrt{8}\right)}\times \frac{\left(\sqrt{9}+\sqrt{8}\right)}{\left(\sqrt{9}+\sqrt{8}\right)}-\frac{1}{\left(\sqrt{8}-\sqrt{7}\right)}\times \frac{\left(\sqrt{8}+\sqrt{7}\right)}{\left(\sqrt{8}+\sqrt{7}\right)}+\frac{1}{\left(\sqrt{7}-\sqrt{6}\right)}\times \frac{\left(\sqrt{7}+\sqrt{6}\right)}{\left(\sqrt{7}+\sqrt{6}\right)}& & & \\ & & & -\frac{1}{\left(\sqrt{6}+\sqrt{5}\right)}\times \frac{\left(\sqrt{6}+\sqrt{5}\right)}{\left(\sqrt{6}+\sqrt{5}\right)}+\frac{1}{\left(\sqrt{5}-\sqrt{4}\right)}\times \frac{\left(\sqrt{5}+\sqrt{4}\right)}{\left(\sqrt{5}+\sqrt{4}\right)}\\ =\frac{\left(\sqrt{9}+\sqrt{8}\right)}{\left(9-8\right)}-\frac{\left(\sqrt{8}+\sqrt{7}\right)}{\left(8-7\right)}+\frac{\left(\sqrt{7}+\sqrt{6}\right)}{\left(7-6\right)}-\frac{\left(\sqrt{6}+\sqrt{5}\right)}{\left(6-5\right)}+\frac{\left(\sqrt{5}+\sqrt{4}\right)}{\left(5-4\right)}& & & \\ =\left(\sqrt{9}+\sqrt{8}\right)-\left(\sqrt{8}+\sqrt{7}\right)+\left(\sqrt{7}+\sqrt{6}\right)-\left(\sqrt{6}+\sqrt{5}\right)+\left(\sqrt{5}+\sqrt{4}\right)=\left(\sqrt{9}+\sqrt{4}\right)=3+2=5.& & & \end{array}$