Rationalize the denominator and simplify the radical expression.
$\(\frac{\sqrt7}{5-\sqrt6}\)$

• 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 that needs rationalization: \( \frac{\sqrt{7}}{5 - \sqrt{6}} \).
2. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator: \( 5 + \sqrt{6} \).
3. Apply the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\) to the denominator, where \(a = 5\) and \(b = \sqrt{6}\).
4. Simplify the denominator: \(5^2 - (\sqrt{6})^2 = 25 - 6 = 19\).
5. Multiply the numerator: \(\sqrt{7} \times (5 + \sqrt{6}) = 5\sqrt{7} + \sqrt{42}\), and combine with the simplified denominator to get \(\frac{5\sqrt{7} + \sqrt{42}}{19}\).