Checkpoint: Rationalizing Denominators with Conjugates Solution

Simplify. Rationalize the denominator.
$\frac{3}{8 - \sqrt{8}}$

$\begin{array}{rcl} \frac{3}{8 - \sqrt{8}} (\frac{8 + \sqrt{8}}{8 + \sqrt{8}}) & = & \frac{24 + 3 \sqrt{8}}{64 - 8} \\ \frac{24 + 3 \sqrt{8}}{56} & = & \frac{24 + 3 (2 \sqrt{2})}{56} \\ \frac{24 + 6 \sqrt{2}}{56} & = & \frac{2 (12 + 3 \sqrt{2})}{56} \\ \frac{12 + 3 \sqrt{2}}{28} \end{array}$

Note

Q: What is the conjugate for this problem?
A: $8 + \sqrt{8}$

Q: Can $\sqrt{8}$ be simplified?

A: Yes, $2 \sqrt{2}$