Question

How do you find fifth roots of 1?

Answer 1

Hint: To find the values of the fifth roots of 1 you need to write the value of 1 as $e^{i2r\pi }$. SO now you can write the value of fifth roots in terms of $e^{i2r\pi }$. Here you take the value of r as 0, 1, 2, 3, 4 to get the five values. $e^{ix}$ can be written as $x\cos x+i\sin x$. Now you can find the different values of x and substitute them to get all the values of the fifth roots of 1.

Complete step by step solution:
The first step we need to do is to write the value of 1 as $e^{i2r\pi }$. So now we can write the value of fifth roots in terms of $e^{i2r\pi }$. Let us take the variable y as the fifth roots of 1.
$\Rightarrow y^5=1$
$\Rightarrow y^5=e^{i2r\pi }$
$\Rightarrow y=e^{i{\displaystyle \frac{2}{5}}r\pi }$
Here the values of r are 0, 1, 2, 3, 4.
We know that $e^{ix}$ can be written as $x\cos x+i\sin x$. Therefore, now we can find the different values of x and substitute them to get all the values of the fifth roots of 1. Here x is
$x={\displaystyle \frac{2}{5}}r\pi$. Therefore we get the values of x as 0 , 72, 144, 216, 288. Therefore, by substituting in this, we get the values as
1, 0.309 + 0.951i , 0.309 – 0.951i , -0.809 + 0.587i, -0.809 - 0.587i.

Therefore, we get the value of fifth root of 1 as 1, 0.309 + 0.951i , 0.309 – 0.951i , -0.809 + 0.587i, -0.809 - 0.587i.

Note: In order to do this question, you need to know the euler's formula. Without this, you cannot solve this problem. You should also know how to write 1 in terms of e. You need to be careful while doing these substitutions.