The square root of -8i is

Options

A. ` +-2(1-i)`
B. `2(1+i)`
C. `+-(1-i)`
D. `+-(1+i)`

Answer

To find the square root of \(-8i\), we can represent it in the form of a complex number \(a + bi\), where \(a\) and \(b\) are real numbers. Here’s a step-by-step solution:

### Step 1: Set up the equation
Let \(z = \sqrt{-8i}\). We can assume \(z\) can be expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ z^2 = -8i \implies (a + bi)^2 = -8i \]
Expanding the left side:
\[ a^2 + 2abi + (bi)^2 = -8i \implies a^2 + 2abi - b^2 = -8i \]
This simplifies to:
\[ (a^2 - b^2) + (2ab)i = -8i \]

### Step 3: Compare real and imaginary parts
From the equation \( (a^2 - b^2) + (2ab)i = 0 - 8i \), we can separate the real and imaginary parts:
1. Real part: \(a^2 - b^2 = 0\)
2. Imaginary part: \(2ab = -8\)

### Step 4: Solve the equations
From the first equation \(a^2 - b^2 = 0\), we can deduce:
\[ a^2 = b^2 \implies a = b \text{ or } a = -b \]

Substituting \(b = a\) into the second equation \(2ab = -8\):
\[ 2a^2 = -8 \implies a^2 = -4 \]
This does not yield real solutions.

Now, substituting \(b = -a\):
\[ 2a(-a) = -8 \implies -2a^2 = -8 \implies a^2 = 4 \implies a = 2 \text{ or } a = -2 \]

### Step 5: Find corresponding \(b\) values
1. If \(a = 2\), then \(b = -2\).
2. If \(a = -2\), then \(b = 2\).

### Step 6: Write the solutions
Thus, we have two solutions:
\[ z = 2 - 2i \quad \text{and} \quad z = -2 + 2i \]

### Step 7: Factor out common terms
We can factor out a 2 from both solutions:
\[ z = 2(1 - i) \quad \text{and} \quad z = -2(1 - i) \]

### Final Result
The square root of \(-8i\) is:
\[ \sqrt{-8i} = \pm 2(1 - i) \]