The number which when multiplied with `(sqrt(3)+sqrt(2))` gives `(sqrt(12)+sqrt(18))`, is

Options

A. `3sqrt(2)-2sqrt(3)`
B. `3sqrt(2)+2sqrt(3)`
C. `sqrt(6)`
D. `2sqrt(3)-3sqrt(2)`

Answer

To solve the problem step by step, we need to find the number \( X \) such that:

\[ X \cdot (\sqrt{3} + \sqrt{2}) = \sqrt{12} + \sqrt{18} \]

### Step 1: Simplify the right-hand side
First, we simplify \( \sqrt{12} \) and \( \sqrt{18} \):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]
\[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

Now, we can rewrite the equation:

\[ X \cdot (\sqrt{3} + \sqrt{2}) = 2\sqrt{3} + 3\sqrt{2} \]

### Step 2: Divide both sides by \( (\sqrt{3} + \sqrt{2}) \)
To isolate \( X \), we divide both sides by \( (\sqrt{3} + \sqrt{2}) \):

\[ X = \frac{2\sqrt{3} + 3\sqrt{2}}{\sqrt{3} + \sqrt{2}} \]

### Step 3: Rationalize the denominator
To simplify this expression, we will multiply the numerator and the denominator by the conjugate of the denominator, which is \( (\sqrt{3} - \sqrt{2}) \):

\[ X = \frac{(2\sqrt{3} + 3\sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]

### Step 4: Simplify the denominator
The denominator simplifies as follows:

\[ (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1 \]

### Step 5: Simplify the numerator
Now, we simplify the numerator:

\[ (2\sqrt{3} + 3\sqrt{2})(\sqrt{3} - \sqrt{2}) = 2\sqrt{3} \cdot \sqrt{3} - 2\sqrt{3} \cdot \sqrt{2} + 3\sqrt{2} \cdot \sqrt{3} - 3\sqrt{2} \cdot \sqrt{2} \]

Calculating each term:

\[ = 2 \cdot 3 - 2\sqrt{6} + 3\sqrt{6} - 3 \cdot 2 \]
\[ = 6 - 3 - 2\sqrt{6} + 3\sqrt{6} \]
\[ = 3 + \sqrt{6} \]

### Step 6: Final expression for \( X \)
Thus, we have:

\[ X = 3 + \sqrt{6} \]

### Conclusion
After evaluating the options, we see that the correct answer is \( \sqrt{6} \) which corresponds to option C.