Chapter 9.6, Problem 35E

To determine

Simplify the expression.

Answer to Problem 35E

The simplified form of expression $\frac{13}{\sqrt{3}+\sqrt{11}}$ is $-\frac{13\sqrt{3}}{8}+\frac{13\sqrt{11}}{8}$ .

Explanation of Solution

Given information:

The expression $\frac{13}{\sqrt{3}+\sqrt{11}}$ .

Formula used:

Rationalizing the denominator.

  $\left(a+b\right)\left(a-b\right)=a^2-b^2$

  $\frac{13}{\sqrt{3}+\sqrt{11}}$

Rationalizing the denominator by multiplying $\sqrt{3}-\sqrt{11}$ in both numerator and denominator,

  $\begin{array}{l}=\frac{13}{\sqrt{3}+\sqrt{11}}\cdot \frac{\sqrt{3}-\sqrt{11}}{\sqrt{3}-\sqrt{11}}\\ \\ =\frac{13\left(\sqrt{3}-\sqrt{11}\right)}{\left(\sqrt{3}+\sqrt{11}\right)\left(\sqrt{3}-\sqrt{11}\right)}\end{array}$

Applying the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in denominator,

  $\begin{array}{l}=\frac{13\sqrt{3}-13\sqrt{11}}{{\left(\sqrt{3}\right)}^2-{\left(\sqrt{11}\right)}^2}\\ \\ =\frac{13\sqrt{3}-13\sqrt{11}}{3-11}\\ \\ =\frac{13\sqrt{3}-13\sqrt{11}}{-8}\\ \\ =-\frac{13\sqrt{3}}{8}+\frac{13\sqrt{11}}{8}\end{array}$

The simplified form of expression $\frac{13}{\sqrt{3}+\sqrt{11}}$ is $=-\frac{13\sqrt{3}}{8}+\frac{13\sqrt{11}}{8}$ .