`(3sqrt(2)+2sqrt(3))/(3sqrt(2)-2sqrt(3))` is equal to

Options

A. a) `5+2sqrt(6)`
B. b) `(3+2sqrt(6))/(2)`
C. c) `5-2sqrt(3)`
D. d) `5+2sqrt(3)`

Answer

To solve the expression \((3\sqrt{2} + 2\sqrt{3}) / (3\sqrt{2} - 2\sqrt{3})\), we will rationalize the denominator. Here are the steps:

### Step-by-Step Solution:

1. **Identify the expression**:
\[ \frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}} \]

2. **Multiply numerator and denominator by the conjugate of the denominator**:
The conjugate of \(3\sqrt{2} - 2\sqrt{3}\) is \(3\sqrt{2} + 2\sqrt{3}\). Thus, we multiply both the numerator and denominator by \(3\sqrt{2} + 2\sqrt{3}\):
\[ \frac{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}{(3\sqrt{2} - 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})} \]

3. **Simplify the denominator using the difference of squares**:
The denominator becomes:
\[ (3\sqrt{2})^2 - (2\sqrt{3})^2 = 9 \cdot 2 - 4 \cdot 3 = 18 - 12 = 6 \]

4. **Expand the numerator**:
The numerator becomes:
\[ (3\sqrt{2})^2 + 2 \cdot (3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 9 \cdot 2 + 12\sqrt{6} + 4 \cdot 3 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6} \]

5. **Combine the results**:
Now, we can write the expression as:
\[ \frac{30 + 12\sqrt{6}}{6} \]

6. **Simplify the fraction**:
Divide both terms in the numerator by 6:
\[ \frac{30}{6} + \frac{12\sqrt{6}}{6} = 5 + 2\sqrt{6} \]

### Final Answer:
Thus, the value of the expression \((3\sqrt{2} + 2\sqrt{3}) / (3\sqrt{2} - 2\sqrt{3})\) is:
\[ \boxed{5 + 2\sqrt{6}} \]