We will rationalize the denominator of \(\frac{\sqrt{6}}{\sqrt{8} + \sqrt{7}}\).
The first step is identifying the radical conjugate. It is \(\sqrt{8} - \sqrt{7}\), because we have a sum of square roots, so the conjugate is a difference of those square roots. Thus we multiply the numerator and denominator by the radical conjugate:
$$\frac{\sqrt{6}(\sqrt{8} - \sqrt{7})}{(\sqrt{8} + \sqrt{7})(\sqrt{8} - \sqrt{7})}$$
Simplify:
$$\frac{\sqrt{48} - \sqrt{42}}{8 - \sqrt{56} + \sqrt{56} - 7} = \frac{4\sqrt{3} - \sqrt{42}}{1} = 4\sqrt{3} - \sqrt{42}$$