Rationalize the denominator and simplify the radical expression.
$\frac{\sqrt{7}}{5-\sqrt{6}}$

• A. $\(\frac{\sqrt7}{19}\)$
• B. $\(\frac{5\sqrt7+\sqrt{42}\)}{-11}$
• C. $\(\frac{5\sqrt7+\sqrt{42}\)}{21}$
• D. $\(\frac{5\sqrt7+\sqrt{42}\)}{19}$

Answer: (D)

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{\sqrt{7}}{5 - \sqrt{6}}\). The goal is to eliminate the radical in the denominator.
2. Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(5 - \sqrt{6}\) is \(5 + \sqrt{6}\). So multiply by \(\frac{5 + \sqrt{6}}{5 + \sqrt{6}}\):
3. \[\frac{\sqrt{7}}{5 - \sqrt{6}} \times \frac{5 + \sqrt{6}}{5 + \sqrt{6}} = \frac{\sqrt{7}(5 + \sqrt{6})}{(5 - \sqrt{6})(5 + \sqrt{6})}\]
4. Use the difference of squares formula for the denominator: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 5\) and \(b = \sqrt{6}\), so the denominator becomes \(5^2 - (\sqrt{6})^2\).
5. Expand the numerator by distributing \(\sqrt{7}\): \(\sqrt{7} \times 5 + \sqrt{7} \times \sqrt{6} = 5\sqrt{7} + \sqrt{42}\). Simplify the denominator by calculating \$25 - 6$.