If `sqrt(x+i y)=+-(a+i b),` then find`sqrt( -x-i ydot)`

Explanation

To solve the problem, we need to find \(\sqrt{-x - i y}\) given that \(\sqrt{x + i y} = \pm (a + i b)\).

### Step-by-step Solution:

1. **Square Both Sides**:
We start with the equation:
\[ \sqrt{x + i y} = \pm (a + i b) \]
Squaring both sides gives:
\[ x + i y = (a + i b)^2 \]

2. **Expand the Right Side**:
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we expand:
\[ (a + i b)^2 = a^2 + 2ab i + (i b)^2 = a^2 + 2ab i - b^2 \]
Thus, we have:
\[ x + i y = (a^2 - b^2) + i(2ab) \]

3. **Compare Real and Imaginary Parts**:
From the equation \(x + i y = (a^2 - b^2) + i(2ab)\), we can equate the real and imaginary parts:
- Real part: \(x = a^2 - b^2\)
- Imaginary part: \(y = 2ab\)

4. **Substitute for \(-x - i y\)**:
We need to find \(\sqrt{-x - i y}\):
\[ -x - i y = - (a^2 - b^2) - i(2ab) \]
This simplifies to:
\[ -x - i y = -a^2 + b^2 - i(2ab) \]

5. **Rearranging the Expression**:
We can rewrite this as:
\[ -x - i y = (b^2 - a^2) - i(2ab) \]

6. **Express as a Square**:
We can recognize this as a perfect square:
\[ (b - ai)^2 = (b^2 + a^2) - 2abi \]
Thus, we can write:
\[ \sqrt{-x - i y} = \sqrt{(b - ai)^2} = \pm (b - ai) \]

### Final Answer:
\[ \sqrt{-x - i y} = \pm (b - ai) \]