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 in the form of \( a + ib \), we will follow these steps:

Given expression:
\[ \frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + i\sqrt{2}) - (\sqrt{3} - i\sqrt{2})} \]

### Step 1: Simplify the Numerator
The numerator can be simplified using the difference of squares formula:
\[ (3 + i\sqrt{5})(3 - i\sqrt{5}) = 3^2 - (i\sqrt{5})^2 \]
Calculating this:
\[ = 9 - (i^2 \cdot 5) = 9 - (-5) = 9 + 5 = 14 \]

### Step 2: Simplify the Denominator
Now, simplify the denominator:
\[ (\sqrt{3} + i\sqrt{2}) - (\sqrt{3} - i\sqrt{2}) = \sqrt{3} + i\sqrt{2} - \sqrt{3} + i\sqrt{2} \]
This simplifies to:
\[ 2i\sqrt{2} \]

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

### Step 4: Simplify the Fraction
We can simplify this fraction:
\[ = \frac{14}{2i\sqrt{2}} = \frac{7}{i\sqrt{2}} \]

### Step 5: Rationalize the Denominator
To express this in the form \( a + ib \), we multiply the numerator and denominator by \( -i \):
\[ = \frac{7 \cdot (-i)}{i\sqrt{2} \cdot (-i)} = \frac{-7i}{-\sqrt{2}} = \frac{7i}{\sqrt{2}} \]

### Step 6: Express in the Form \( a + ib \)
This can be expressed as:
\[ 0 + i\frac{7}{\sqrt{2}} \]

Thus, we have:
\[ a = 0, \quad b = \frac{7}{\sqrt{2}} \]

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