√i = ?

Steps to Solve

1. Express the imaginary unit 'i' in exponential form

The imaginary unit can be expressed using Euler's formula. Recall that:
$$ i = e^{i \frac{\pi}{2}} $$

2. Apply the square root formula in exponential form

To find the square root of 'i', we use the square root property for complex exponentials:
$$ \sqrt{z} = e^{\frac{1}{2} \log(z)} $$
So, we have:
$$ \sqrt{i} = \sqrt{e^{i \frac{\pi}{2}}} = e^{i \frac{\pi}{4}} $$

3. Convert back to rectangular form

Now, we convert $e^{i \frac{\pi}{4}}$ back into rectangular form using Euler's formula:
$$ e^{i \theta} = \cos(\theta) + i \sin(\theta) $$
Thus:
$$ \sqrt{i} = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) $$

4. Calculate the trigonometric values

The values for cosine and sine at $\frac{\pi}{4}$ are both equal to $\frac{\sqrt{2}}{2}$:
$$ \sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} $$

5. Final presentation

We can also write the answer in a more concise form for clarity:
$$ \sqrt{i} = \frac{\sqrt{2}}{2}(1 + i) $$