# Thread: 4th roots of i in a +bi form

1. ## 4th roots of i in a +bi form

Find the fourth roots of i in a +bi form. Hint: the half angle formula may be helpful.

Can anyone help? Thanks!

2. Originally Posted by reastland
Find the fourth roots of i in a +bi form. Hint: the half angle formula may be helpful.
First $\displaystyle i = \exp \left( {\frac{{\pi i}}{2}} \right)$ so that one fourth root is $\displaystyle \exp \left( {\frac{{\pi i}}{8}} \right).$

What are the other three?

3. It also helps to remember that there will be four fourth-roots, and they are all evenly spaced around a circle...

4. Thanks. I don't follow what you mean by "exp". Can you explain that a little further?

5. exp means e = 2.718.....

6. In other words, $\displaystyle \displaystyle \exp{\left(\frac{\pi i}{2}\right)} = e^{\frac{\pi i}{2}}$.

7. Originally Posted by reastland
Thanks. I don't follow what you mean by "exp". Can you explain that a little further?
If you are not familiar with the notation $\displaystyle \exp(x+yi)$ here is an elementary way to use it.
Think of $\displaystyle \exp(x+yi)$ as $\displaystyle r(\cos(x)+i~\sin(y))$ where $\displaystyle r=\sqrt{x^2+y^2}$.

Example: $\displaystyle \exp(3+4i)=5(\cos(\theta)+i~\sin(\theta)$ where $\displaystyle \theta=\arctan \left(\frac{4}{3}\right)$.