Complex Numbers Help ASAP

Question

I just need help rewriting a complex number $(81i)^{\frac 14}$ in the form $re^{iθ}$. An explanation of how to do this would be amazing, thank you.

Answer 1

So you are asking for the roots of $(81i)^\dfrac{1}{4}$

We know that the angle to the $x$-axis is $90$ degrees.

The first solution therefore is $\alpha_1 = \dfrac{90}{4}$ degrees

The other 3 solutions will therefore be:

$\alpha_n = \alpha_1 + (n-1) * 90 $

The length is simply $81^\dfrac{1}{4} = 3$

Your result is:

$$3 e^{\alpha_n i}$$

Answer 2

$(3^4e^{i\pi/2})^{1/4}=3e^{i(\pi/8+n2\pi/4)}$ with $n\in\{0,\,1,\,2,\,3\}$.

Answer 3

So the first remark that has to be given here is that $(81i)^\tfrac{1}{4}$ is not one number but actually four numbers.

The way to solve this problem is by either looking up the formula for complex roots or deriving it by hand:

First lets rewrite $81i$ in polar form: $81i=3^3\cdot e^{\pi i/2}$.

Now we look for complex numbers $z=r\cdot e^{i\varphi}$ such that: $z^4=r^4e^{4i\varphi}=3^4\cdot e^{\pi i/2+k2\pi i}$ for some integer $k$. Here we used the standard formula for multiplication in polar form and the fact that $e^{k2\pi i}=1$. Comparing yields: $r=3$ and $\varphi=\pi/8+k\pi/2$. Thus you have 4 different solutions namely: $z_k=3\cdot e^{\pi i/8+k\pi i/2}$ for $k\in\{0,1,2,3\}$.

Hope that helps.