Evaluate:
(iv) `((3+sqrt5i)(3-sqrt5i))/((sqrt3+sqrt2i)-(sqrt3-sqrt2i))`

Answer

To evaluate the expression \(\frac{(3+\sqrt{5}i)(3-\sqrt{5}i)}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-\sqrt{2}i)}\), we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \((3+\sqrt{5}i)(3-\sqrt{5}i)\). This is a product of conjugates, which can be simplified using the identity \(a^2 - b^2\).

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

Calculating \(3^2\) and \((\sqrt{5}i)^2\):

\[ = 9 - (5i^2) \]
Since \(i^2 = -1\), we have:

\[ = 9 - 5(-1) = 9 + 5 = 14 \]

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

\[ = \sqrt{3} + \sqrt{2}i - \sqrt{3} + \sqrt{2}i = 2\sqrt{2}i \]

### 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
We can simplify \(\frac{14}{2\sqrt{2}i}\):

\[ = \frac{14}{2\sqrt{2}} \cdot \frac{1}{i} = \frac{7}{\sqrt{2}} \cdot \frac{1}{i} \]

To eliminate \(i\) from the denominator, we multiply the numerator and denominator by \(i\):

\[ = \frac{7i}{\sqrt{2}i^2} \]

Since \(i^2 = -1\):

\[ = \frac{7i}{-\sqrt{2}} = -\frac{7i}{\sqrt{2}} \]

### Final Answer
Thus, the evaluated expression is:

\[ -\frac{7i}{\sqrt{2}} \]

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