Question

what is the following quotient?

\frac{9+\sqrt{2}}{4-\sqrt{7}}

\circ \frac{9\sqrt{7}+\sqrt{14}}{-3}

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

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

\circ \frac{79}{9}

Explanation:

Identify the conjugate of the denominator

Using the Radical Conjugates knowledge point

$$ \text{Conjugate of } 4 - \sqrt{7} \text{ is } 4 + \sqrt{7} $$

Multiply the numerator and denominator by the conjugate

Using the Rationalizing the Denominator knowledge point

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

Expand the numerator

Using the Multiplying Binomial Radicals knowledge point

$$ (9 + \sqrt{2})(4 + \sqrt{7}) = 9(4) + 9(\sqrt{7}) + \sqrt{2}(4) + \sqrt{2}(\sqrt{7}) = 36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14} $$

Simplify the denominator

Using the Radical Conjugates knowledge point

$$ (4 - \sqrt{7})(4 + \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9 $$

Write the final rationalized expression

Combine the simplified numerator and denominator:

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

Answer:

• (A) \(\frac{9\sqrt{7} + \sqrt{14}}{-3}\)
• (B) \(\frac{36 - 9\sqrt{7} + 4\sqrt{2} - \sqrt{14}}{9}\)
• (C) \(\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}\) (Correct answer)
• (D) \(\frac{79}{9}\)