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.
### 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.
