Express the following in the form a+bi, where a and b are real numbers
`((3 + sqrt5i) (3-sqrt5i))/((sqrt3+sqrt2i)- (sqrt3-sqrt2i))`

Answer

To express the given expression \(\frac{(3 + \sqrt{5}i)(3 - \sqrt{5}i)}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - \sqrt{2}i)}\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers, we can follow these steps:

### Step 1: Simplify the Denominator
The denominator is \((\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - \sqrt{2}i)\). We can simplify this:
\[ \sqrt{3} + \sqrt{2}i - \sqrt{3} + \sqrt{2}i = 2\sqrt{2}i \]

### Step 2: Simplify the Numerator
The numerator is \((3 + \sqrt{5}i)(3 - \sqrt{5}i)\). We can use the difference of squares formula:
\[ (3 + \sqrt{5}i)(3 - \sqrt{5}i) = 3^2 - (\sqrt{5}i)^2 = 9 - 5(-1) = 9 + 5 = 14 \]

### 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: Rationalize the Denominator
To express this in the form \(a + bi\), we can multiply the numerator and denominator by \(-i\):
\[ \frac{14 \cdot (-i)}{2\sqrt{2}i \cdot (-i)} = \frac{-14i}{2\sqrt{2}(-1)} = \frac{-14i}{-2\sqrt{2}} = \frac{14i}{2\sqrt{2}} = \frac{7i}{\sqrt{2}} \]

### Step 5: Express in the Form \(a + bi\)
This can be rewritten as:
\[ 0 + \frac{7}{\sqrt{2}}i \]
Thus, \(a = 0\) and \(b = \frac{7}{\sqrt{2}}\).

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