Question

what is the following quotient?
\\(\frac{9 + sqrt{2}}{4 - sqrt{7}}\\)
\\(\frac{9sqrt{7} + sqrt{14}}{-3}\\)
\\(\frac{36 - 9sqrt{7} + 4sqrt{2} - sqrt{14}}{9}\\)
\\(\frac{36 + 9sqrt{7} + 4sqrt{2} + sqrt{14}}{9}\\)
\\(\frac{79}{9}\\)

Explanation:

Step1: Rationalize the Denominator

To simplify \(\frac{9 + \sqrt{2}}{4 - \sqrt{7}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(4 + \sqrt{7}\).

$$ \frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})} $$

Step2: Expand the Numerator

Using the distributive property (FOIL method) for the numerator:

$$
LATEXBLOCK0
$$

Step3: Expand the Denominator

Using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\) for the denominator:

$$
LATEXBLOCK1
$$

Step4: Combine Numerator and Denominator

Putting the expanded numerator and denominator together, we get:

$$ \frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9} $$

Answer:

\(\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}\) (the third option)