If `a=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` and `b=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))` , then find the value of `3(a^(2)-b^(2))` .

Answer

To solve the problem, we need to find the value of \( 3(a^2 - b^2) \) given:

\[ a = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \quad \text{and} \quad b = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \]

### Step 1: Use the identity for \( a^2 - b^2 \)

We can use the identity \( a^2 - b^2 = (a+b)(a-b) \). Thus, we need to find \( a + b \) and \( a - b \).

### Step 2: Calculate \( a + b \)

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

To add these fractions, we find a common denominator:

\[ a + b = \frac{(\sqrt{3} + \sqrt{2})^2 + (\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 3: Expand the numerators

Calculating the squares:

\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6} \]
\[ (\sqrt{3} - \sqrt{2})^2 = 3 + 2 - 2\sqrt{6} = 5 - 2\sqrt{6} \]

Now, add these results:

\[ a + b = \frac{(5 + 2\sqrt{6}) + (5 - 2\sqrt{6})}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{10}{3 - 2} = \frac{10}{1} = 10 \]

### Step 4: Calculate \( a - b \)

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

Using a similar approach as before, we find a common denominator:

\[ a - b = \frac{(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 5: Expand the numerators for \( a - b \)

Using the difference of squares:

\[ (\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2 = (5 + 2\sqrt{6}) - (5 - 2\sqrt{6}) = 4\sqrt{6} \]

Thus,

\[ a - b = \frac{4\sqrt{6}}{1} = 4\sqrt{6} \]

### Step 6: Calculate \( a^2 - b^2 \)

Now we can substitute back into the identity:

\[ a^2 - b^2 = (a + b)(a - b) = 10 \cdot 4\sqrt{6} = 40\sqrt{6} \]

### Step 7: Find \( 3(a^2 - b^2) \)

Finally, we multiply by 3:

\[ 3(a^2 - b^2) = 3 \cdot 40\sqrt{6} = 120\sqrt{6} \]

### Final Answer

Thus, the value of \( 3(a^2 - b^2) \) is:

\[ \boxed{120\sqrt{6}} \]