Chapter 9.1, Problem 43E

To determine

To write: A factor to rationalize the denominator of the expression $\frac{\sqrt{2}}{\sqrt{5}-8}$

Answer to Problem 43E

The required answer is: factor to rationalize the denominator is $\sqrt{5}+8$

Explanation of Solution

Given information:

The given expression: $\frac{\sqrt{2}}{\sqrt{5}-8}$

Calculation:

  $\frac{\sqrt{2}}{\sqrt{5}-8}=\frac{\sqrt{2}}{\sqrt{5}-8}\cdot \frac{\sqrt{5}+8}{\sqrt{5}+8}$

(The conjugate of $\sqrt{5}-8$ is $\sqrt{5}+8$ )

  $=\frac{\sqrt{2}(\sqrt{5}+8)}{{(\sqrt{5})}^2-{(8)}^2}$

(Sum and difference pattern)

  $\begin{array}{l}=\frac{\sqrt{10}+8\sqrt{2}}{5-64}\\ =\frac{\sqrt{10}+8\sqrt{2}}{-59}\end{array}$

(Simplify)