What is the value of `sqrt(-i)`, where `i = sqrt( -1)` ?

Options

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

Answer

To find the value of \(\sqrt{-i}\), where \(i = \sqrt{-1}\), we can follow these steps:

### Step 1: Rewrite \(-i\) in exponential form
We know that \(i\) can be expressed in polar form as:
\[ i = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) \]
Thus, \(-i\) can be written as:
\[ -i = \cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right) \]

### Step 2: Use Euler's formula
Using Euler's formula, we can express \(-i\) as:
\[ -i = e^{-i\frac{\pi}{2}} \]

### Step 3: Take the square root
Now, we take the square root of both sides:
\[ \sqrt{-i} = \sqrt{e^{-i\frac{\pi}{2}}} \]
This can be simplified using the property of exponents:
\[ \sqrt{e^{-i\frac{\pi}{2}}} = e^{-i\frac{\pi}{4}} \]

### Step 4: Convert back to rectangular form
Using Euler's formula again, we can convert back to rectangular form:
\[ e^{-i\frac{\pi}{4}} = \cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right) \]
Calculating the cosine and sine values:
\[ \cos\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]
Thus, we have:
\[ \sqrt{-i} = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \]

### Step 5: Final expression
We can express this as:
\[ \sqrt{-i} = \frac{1 - i}{\sqrt{2}} \]

### Conclusion
The value of \(\sqrt{-i}\) is:
\[ \sqrt{-i} = \pm \frac{1 - i}{\sqrt{2}} \]