What is the following quotient?

[tex]\[ \frac{9+\sqrt{2}}{4-\sqrt{7}} \][/tex]

A. \(\frac{9 \sqrt{7}+\sqrt{14}}{-3}\)

B. \(\frac{36-9 \sqrt{7}+4 \sqrt{2}-\sqrt{14}}{9}\)

C. \(\frac{36+9 \sqrt{7}+4 \sqrt{2}+\sqrt{14}}{9}\)

D. [tex]\(\frac{79}{9}\)[/tex]



Answer :

When solving for the following quotients, we break each expression down step-by-step, applying algebraic manipulations as needed.

### 1. Expression:
[tex]\[ \frac{9 + \sqrt{2}}{4 - \sqrt{7}} \][/tex]

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \(4 + \sqrt{7}\):
[tex]\[ \frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})} \][/tex]

### 2. Expression:
[tex]\[ \frac{9\sqrt{7} + \sqrt{14}}{-3} \][/tex]

Simplify the expression by dividing each term in the numerator by \(-3\):
[tex]\[ \frac{9\sqrt{7}}{-3} + \frac{\sqrt{14}}{-3} = -3\sqrt{7} - \frac{\sqrt{14}}{3} \][/tex]

This yields:
[tex]\[ -3\sqrt{7} - \frac{\sqrt{14}}{3} \][/tex]

### 3. Expression:
[tex]\[ \frac{36 - 9\sqrt{7} + 4\sqrt{2} - \sqrt{14}}{9} \][/tex]

Separate each term in the numerator:
[tex]\[ \frac{36}{9} - \frac{9\sqrt{7}}{9} + \frac{4\sqrt{2}}{9} - \frac{\sqrt{14}}{9} \][/tex]

This simplifies to:
[tex]\[ 4 - \sqrt{7} + \frac{4\sqrt{2}}{9} - \frac{\sqrt{14}}{9} \][/tex]

Combining all terms, we obtain:
[tex]\[ -\sqrt{7} + 4 - \frac{\sqrt{14}}{9} + \frac{4\sqrt{2}}{9} \][/tex]

### 4. Expression:
[tex]\[ \frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9} \][/tex]

Separate each term in the numerator:
[tex]\[ \frac{36}{9} + \frac{9\sqrt{7}}{9} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} \][/tex]

This simplifies to:
[tex]\[ 4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} \][/tex]

Combining all terms, we obtain:
[tex]\[ \sqrt{7} + 4 + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9} \][/tex]

### 5. Expression:
[tex]\[ \frac{79}{9} \][/tex]

This is already in its simplest form and does not require any further simplification.

### Summary of Results:
1. \(\frac{9 + \sqrt{2}}{4 - \sqrt{7}}\) remains in its complex form since rationalization is not needed for this context.
2. \(-3\sqrt{7} - \frac{\sqrt{14}}{3}\)
3. \(-\sqrt{7} + 4 - \frac{\sqrt{14}}{9} + \frac{4\sqrt{2}}{9}\)
4. \(\sqrt{7} + 4 + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9}\)
5. \(\frac{79}{9}\)

Each part simplifies according to the steps shown to reveal their final forms.