If ` sqrt( x + iy) = pm (a + ib) , " then " sqrt( - x - iy)` is equal to

Options

A. `pm (b +ia)`
B. `pm (a - ib)`
C. `pm (b - ia)`
D. none of these

Answer

To solve the problem, we start with the equation given:

\[ \sqrt{x + iy} = \pm (a + ib) \]

### Step 1: Square Both Sides
We square both sides to eliminate the square root:

\[ x + iy = (a + ib)^2 \]

### Step 2: Expand the Right Side
Now we expand the right side:

\[ (a + ib)^2 = a^2 + 2iab + (ib)^2 = a^2 + 2iab - b^2 \]

Thus, we can write:

\[ x + iy = (a^2 - b^2) + i(2ab) \]

### Step 3: Equate Real and Imaginary Parts
From the above equation, we can equate the real and imaginary parts:

1. Real part: \( x = a^2 - b^2 \)
2. Imaginary part: \( y = 2ab \)

### Step 4: Find \(-x - iy\)
Now we need to find \(\sqrt{-x - iy}\). We can express \(-x - iy\) using the values we found:

\[ -x - iy = -(a^2 - b^2) - i(2ab) = -a^2 + b^2 - i(2ab) \]

### Step 5: Rewrite in Standard Form
This can be rewritten as:

\[ (-a^2 + b^2) - i(2ab) \]

### Step 6: Find the Square Root
Now we need to find the square root of this expression. We can express it as:

\[ \sqrt{(-a^2 + b^2) - i(2ab)} \]

Using the property of square roots of complex numbers, we can write:

\[ \sqrt{(-a^2 + b^2) - i(2ab)} = \pm (c + id) \]

where \(c\) and \(d\) need to be determined.

### Step 7: Equate Again
We can set:

\[ c^2 - d^2 = -a^2 + b^2 \]
\[ 2cd = -2ab \]

### Step 8: Solve for \(c\) and \(d\)
From \(2cd = -2ab\), we can simplify to:

\[ cd = -ab \]

Now we can solve the equations for \(c\) and \(d\).

### Final Result
After solving, we find that:

\[ \sqrt{-x - iy} = \pm (b - ai) \]

### Summary
Thus, the final answer is:

\[ \sqrt{-x - iy} = \pm (b - ai) \]