Express the following expression in the form of `a + i b``((3+isqrt(5))(3-isqrt(5)))/((sqrt(3)+sqrt(2)i)-(sqrt(3)-isqrt(2)))`

Explanation

To express the given expression \(\frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})}\) in the form \(a + ib\), we can follow these steps:

### Step 1: Simplify the Numerator
The numerator is \((3 + i\sqrt{5})(3 - i\sqrt{5})\). This is in the form of \( (a + b)(a - b) = a^2 - b^2 \).

\[ = 3^2 - (i\sqrt{5})^2 \]
\[ = 9 - (i^2 \cdot 5) \]
Since \(i^2 = -1\), we have:
\[ = 9 - (-5) = 9 + 5 = 14 \]

### Step 2: Simplify the Denominator
The denominator is \((\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})\).

Distributing the negative sign:
\[ = \sqrt{3} + \sqrt{2}i - \sqrt{3} + i\sqrt{2} \]
The \(\sqrt{3}\) terms cancel out:
\[ = \sqrt{2}i + \sqrt{2}i = 2\sqrt{2}i \]

### Step 3: Combine the Results
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{14}{2\sqrt{2}i} \]

### Step 4: Simplify the Fraction
To simplify \(\frac{14}{2\sqrt{2}i}\):
\[ = \frac{14}{2\sqrt{2}} \cdot \frac{1}{i} \]
\[ = \frac{7}{\sqrt{2}} \cdot \frac{-i}{-1} = -\frac{7i}{\sqrt{2}} \]

### Step 5: Express in the Form \(a + ib\)
We can express \(-\frac{7i}{\sqrt{2}}\) as:
\[ 0 + i\left(-\frac{7}{\sqrt{2}}\right) \]
Thus, we have \(a = 0\) and \(b = -\frac{7}{\sqrt{2}}\).

### Final Answer
The expression in the form \(a + ib\) is:
\[ 0 + i\left(-\frac{7}{\sqrt{2}}\right) \]