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$.