Chapter 9.3, Problem 9Q

To determine

To Simplify: The expression 13

  $\frac{6}{5+\sqrt{3}}$

Answer to Problem 9Q

The required answer is $\frac{3\left(5-\sqrt{3}\right)}{11}$ .

Explanation of Solution

Given information:

The expression $\frac{6}{5+\sqrt{3}}$

Concept Used:

A conjugate is when the sign in the middle of two term is changed.

For example:

  $\begin{array}{l}\text{The conjugate of }4x+1\text{ is }4x-1\\ \text{The conjugate of }a-\sqrt{b}\text{ is }a+\sqrt{b}\\ \text{Where }a,b\text{ are rational numbers}\text{. }\end{array}$

Calculation:

  $\frac{6}{5+\sqrt{3}}$

Rationalize the denominator

The conjugate of $5+\sqrt{3}$ is $5-\sqrt{3}$

  $\begin{array}{l}=\frac{6}{5+\sqrt{3}}\cdot \frac{5-\sqrt{3}}{5-\sqrt{3}}\\ =\frac{6\left(5-\sqrt{3}\right)}{5^2-{\left(\sqrt{3}\right)}^2}\\ =\frac{6\left(5-\sqrt{3}\right)}{25-3}\\ =\frac{6\left(5-\sqrt{3}\right)}{22}\\ =\frac{3\left(5-\sqrt{3}\right)}{11}\end{array}$