Chapter 9.6, Problem 34E

To determine

Simplify the expression.

Answer to Problem 34E

The simplified form of expression $\frac{8}{\sqrt{10}+3}$ is $8\sqrt{10}-24$ .

Explanation of Solution

Given information:

The expression $\frac{8}{\sqrt{10}+3}$ .

Formula used:

Rationalizing the denominator.

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

  $\frac{8}{\sqrt{10}+3}$

Rationalizing the denominator by multiplying $\sqrt{10}-3$ in both numerator and denominator,

  $=\frac{8}{\sqrt{10}+3}\cdot \frac{\sqrt{10}-3}{\sqrt{10}-3}$

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

  $\begin{array}{l}=\frac{8\left(\sqrt{10}-3\right)}{{\left(\sqrt{10}\right)}^2-{\left(3\right)}^2}\\ \\ =\frac{8\left(\sqrt{10}-3\right)}{10-9}\\ \\ =\frac{8\sqrt{10}-24}{1}\\ \\ =8\sqrt{10}-24\end{array}$

The simplified form of expression $\frac{8}{\sqrt{10}+3}$ is $8\sqrt{10}-24$ .