If `a=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` and `b=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))`, then what is the value of `(a^(2)+b^(2)-ab)`?

Options

A. `97`
B. `(2sqrt(3))+2`
C. `(4sqrt(6))+1`
D. `98`

Answer

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

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

### Step 1: Find the product \( ab \)

First, let's calculate \( ab \):

\[ ab = \left( \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \right) \left( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \right) \]

Notice that the terms in the numerator and denominator will cancel out:

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

### Step 2: Express \( b \) in terms of \( a \)

Since \( ab = 1 \), we can express \( b \) as:

\[ b = \frac{1}{a} \]

### Step 3: Calculate \( a^2 + b^2 \)

Using the expression for \( b \):

\[ b^2 = \left(\frac{1}{a}\right)^2 = \frac{1}{a^2} \]

Now, we can find \( a^2 + b^2 \):

\[ a^2 + b^2 = a^2 + \frac{1}{a^2} \]

### Step 4: Use the identity for \( a^2 + b^2 \)

We can use the identity:

\[ a^2 + b^2 = (a + b)^2 - 2ab \]

Since \( ab = 1 \), we have:

\[ a^2 + b^2 = (a + b)^2 - 2 \]

### Step 5: Calculate \( a + b \)

Now we need to find \( a + b \):

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

To combine these fractions, we find a common denominator:

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

Calculating the numerator:

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

Adding these:

\[ (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10 \]

Now, the denominator:

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

Thus:

\[ a + b = 10 \]

### Step 6: Substitute back to find \( a^2 + b^2 \)

Now substitute \( a + b \) back into our earlier expression:

\[ a^2 + b^2 = (10)^2 - 2 = 100 - 2 = 98 \]

### Step 7: Calculate \( a^2 + b^2 - ab \)

Finally, we find:

\[ a^2 + b^2 - ab = 98 - 1 = 97 \]

### Final Answer

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

\[ \boxed{97} \]