`sqrt(2i)` equals

Options

A. `1 + i`
B. `1 -i`
C. `- sqrt(2) i`
D. none of these

Answer

To find the value of \(\sqrt{2i}\), we can follow these steps:

### Step 1: Express \(2i\) in polar form
First, we need to express \(2i\) in polar form. The complex number \(2i\) can be written as:
\[ 2i = 0 + 2i \]
The modulus \(r\) is given by:
\[ r = |2i| = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]
The argument \(\theta\) is:
\[ \theta = \tan^{-1}\left(\frac{2}{0}\right) = \frac{\pi}{2} \]
Thus, in polar form, we have:
\[ 2i = 2\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) \]

### Step 2: Apply the square root in polar form
To find the square root of a complex number in polar form, we use:
\[ \sqrt{r} \left(\cos\frac{\theta}{2} + i\sin\frac{\theta}{2}\right) \]
For \(2i\):
- The modulus \(r = 2\) gives \(\sqrt{r} = \sqrt{2}\).
- The argument \(\theta = \frac{\pi}{2}\) gives \(\frac{\theta}{2} = \frac{\pi}{4}\).

Thus, we have:
\[ \sqrt{2i} = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \]

### Step 3: Calculate \(\cos\frac{\pi}{4}\) and \(\sin\frac{\pi}{4}\)
We know that:
\[ \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} \]
So we can substitute these values:
\[ \sqrt{2i} = \sqrt{2}\left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) \]

### Step 4: Simplify the expression
Now, simplifying this expression:
\[ \sqrt{2i} = \sqrt{2} \cdot \frac{1}{\sqrt{2}} + i\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 + i \]

### Conclusion
Thus, we find that:
\[ \sqrt{2i} = 1 + i \]

### Final Answer
The correct option is \(A: 1 + i\).

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