Question

rationalize the denominator.
\\(\frac{9}{2 - \sqrt{7}}\\)
\\(\frac{9}{2 - \sqrt{7}} = \square\\)
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Identify the conjugate of the denominator

The denominator is \(2 - \sqrt{7}\), so its conjugate is \(2 + \sqrt{7}\). We multiply the numerator and the denominator by this conjugate to rationalize the denominator.
\[
\frac{9}{2 - \sqrt{7}} \times \frac{2 + \sqrt{7}}{2 + \sqrt{7}}
\]

Step2: Multiply the numerators and the denominators

First, multiply the numerators: \(9\times(2 + \sqrt{7}) = 18 + 9\sqrt{7}\)
Then, multiply the denominators using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 2\) and \(b=\sqrt{7}\). So, \((2 - \sqrt{7})(2 + \sqrt{7})=2^2-(\sqrt{7})^2 = 4 - 7=-3\)
Now we have the fraction \(\frac{18 + 9\sqrt{7}}{-3}\)

Step3: Simplify the fraction

Divide each term in the numerator by \(-3\):
\[
\frac{18}{-3}+\frac{9\sqrt{7}}{-3}=-6 - 3\sqrt{7}
\]

Answer:

\(-6 - 3\sqrt{7}\)