Chapter 6, Problem 12CT

To determine

To Simplify:

  $\frac{2}{3-\sqrt{5}}$ .

Answer to Problem 12CT

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

Explanation of Solution

Given :

The expression is :

  $\frac{2}{3-\sqrt{5}}$

Calculation:

Given:

  $\begin{array}{l}\frac{2}{3-\sqrt{5}}\\ =\frac{2}{3-\sqrt{5}}\cdot \frac{3+\sqrt{5}}{3+\sqrt{5}}\mathrm{............}\left[\begin{array}{l}\text{rationalize the denominator by multiplying and dividing}\\ by\text{ the conjugate of the denominator}\text{.}\end{array}\right]\\ =\frac{2\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\\ =\frac{2\left(3+\sqrt{5}\right)}{{\left(3\right)}^2-{\left(\sqrt{5}\right)}^2}\mathrm{...............}[\left(a+b\right)\left(a-b\right)=a^2-b^2]\\ =\frac{2\left(3+\sqrt{5}\right)}{9-5}\\ =\frac{2\left(3+\sqrt{5}\right)}{4}\\ =\frac{3+\sqrt{5}}{2}\\ so,\\ \frac{2}{3-\sqrt{5}}=\frac{3+\sqrt{5}}{2}\end{array}$