Multiplicative inverse of a `(-3)/(sqrt(3) - sqrt(2))` is

Options

A. `3/(sqrt(3) - sqrt(2))`
B. `-(1)/(3(sqrt(3) + sqrt(2)))`
C. `(sqrt(3) + sqrt(2))/(3)`
D. `1`

Answer

To find the multiplicative inverse of the expression \(-\frac{3}{\sqrt{3} - \sqrt{2}}\), we can follow these steps:

### Step 1: Understand the Expression
The expression we need to find the multiplicative inverse for is:
\[ Z = -\frac{3}{\sqrt{3} - \sqrt{2}} \]

### Step 2: Find the Multiplicative Inverse
The multiplicative inverse of a number \(Z\) is given by:
\[ Z^{-1} = \frac{1}{Z} \]
Thus, we have:
\[ Z^{-1} = \frac{1}{-\frac{3}{\sqrt{3} - \sqrt{2}}} \]

### Step 3: Simplify the Expression
To simplify \(\frac{1}{-\frac{3}{\sqrt{3} - \sqrt{2}}}\), we can multiply the numerator and the denominator by \(\sqrt{3} + \sqrt{2}\) (the conjugate of the denominator):
\[ Z^{-1} = \frac{\sqrt{3} + \sqrt{2}}{-3} \]

### Step 4: Final Expression
Thus, the multiplicative inverse can be expressed as:
\[ Z^{-1} = -\frac{\sqrt{3} + \sqrt{2}}{3} \]

### Conclusion
The multiplicative inverse of \(-\frac{3}{\sqrt{3} - \sqrt{2}}\) is:
\[ -\frac{\sqrt{3} + \sqrt{2}}{3} \]