Find the values of a and b in each of the
`(3)/(sqrt(3)-sqrt(2))= a sqrt(3)-bsqrt(2)`

Answer

To solve the equation \(\frac{3}{\sqrt{3}-\sqrt{2}} = a\sqrt{3} - b\sqrt{2}\), we will follow these steps:

### Step 1: Rationalize the Denominator
To eliminate the square roots in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{3} - \sqrt{2}\) is \(\sqrt{3} + \sqrt{2}\).

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

### Step 2: Simplify the Denominator
Using the difference of squares formula, we can simplify the denominator:

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

So, the denominator simplifies to 1.

### Step 3: Simplify the Numerator
Now, we simplify the numerator:

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

### Step 4: Set the Equation
Now we have:

\[ 3\sqrt{3} + 3\sqrt{2} = a\sqrt{3} - b\sqrt{2} \]

### Step 5: Compare Coefficients
We can compare the coefficients of \(\sqrt{3}\) and \(\sqrt{2}\) from both sides of the equation.

1. For \(\sqrt{3}\):
\[ a = 3 \]

2. For \(\sqrt{2}\):
\[ -b = 3 \implies b = -3 \]

### Final Values
Thus, the values of \(a\) and \(b\) are:
\[ a = 3, \quad b = -3 \]

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