Find the square roots of : 4ab - `2i( a^(2) - b^(2)) `

Answer

To find the square roots of the expression \(4ab - 2i(a^2 - b^2)\), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
Start with the given expression:
\[ 4ab - 2i(a^2 - b^2) \]
We can rewrite \(a^2 - b^2\) using the difference of squares:
\[ a^2 - b^2 = (a + b)(a - b) \]
Thus, we can express the original expression as:
\[ 4ab - 2i(a + b)(a - b) \]

2. **Recognize the Structure**:
Notice that \(4ab\) can be rewritten in terms of squares:
\[ 4ab = (a + b)^2 - (a^2 + b^2) \]
However, for our purpose, we will keep it as \(4ab\) for now and focus on the entire expression:
\[ 4ab - 2i(a + b)(a - b) \]

3. **Combine Terms**:
We can express the entire expression as:
\[ 4ab - 2i(a + b)(a - b) = (a + b)^2 + i(a - b)^2 \]
This is because:
\[ (a + b)^2 = a^2 + 2ab + b^2 \quad \text{and} \quad (a - b)^2 = a^2 - 2ab + b^2 \]

4. **Finding the Square Root**:
We need to find the square root of the expression:
\[ (a + b)^2 + i(a - b)^2 \]
This can be expressed in the form of a complex number:
\[ z = (a + b) + i(a - b) \]
Therefore, the square root can be expressed as:
\[ \sqrt{z} = \pm \left( (a + b) + i(a - b) \right) \]

5. **Final Result**:
Thus, the square roots of the expression \(4ab - 2i(a^2 - b^2)\) are:
\[ \pm \left( (a + b) - i(a - b) \right) \]