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 denominator contains a radical, so we want to eliminate it by multiplying by the conjugate.
2. Write the conjugate of the denominator, which is \$5 + \sqrt{6}\(, and multiply both numerator and denominator by this conjugate to keep the expression equivalent: \)\frac{\sqrt{7}}{5 - \sqrt{6}} \times \frac{5 + \sqrt{6}}{5 + \sqrt{6}}$.
3. Multiply the numerators: \(\sqrt{7} \times (5 + \sqrt{6}) = 5\sqrt{7} + \sqrt{7} \times \sqrt{6} = 5\sqrt{7} + \sqrt{42}\).
4. Multiply the denominators using the difference of squares formula: \((5 - \sqrt{6})(5 + \sqrt{6}) = 5^2 - (\sqrt{6})^2 = 25 - 6 = 19\).
5. Write the new expression as \(\frac{5\sqrt{7} + \sqrt{42}}{19}\). This is the rationalized and simplified form of the original expression.