## root of unity

Dear Dr Kim

I know you must be recieving many emails of this sort at this moment in time, so i’ll try and keep this as brief as possible: just a quick (easy) question,

When trying to calculate the roots of a polynomial say

$t^3 -5$

I know it can be split into its linear factors where $a$= cube root of 5

$(t-a)(t-aw)(t-aw^2)$

and $w=exp(2\pi/3)$.

How do you calculate the value of $w$?

many thanks
sorry again for the inconvenience

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Importantly, the correct value is $w=exp(2\pi i/3)$ not $exp(2\pi/3)$. This is the complex number on the unit circle at an angle of $2\pi/3$ with the positive real axis. $exp(2\pi/3)$, on the other hand, is a real number. $1,w$, and $w^2$ are the three solutions to the equation

$t^3=1.$

For this reason, whenever $\alpha$ is a solution to an equation of the form

$t^3-z=0,$

so are $\alpha w$ and $\alpha w^2$.

As for the value, it depends on the problem. Sometimes, $exp(2\pi i/3)$ is a perfectly legitimate final expression for a number. On the other hand, you can write it in terms of radicals as

$exp(2\pi i/3)=cos(2\pi/3)+isin(2\pi/3)=-1/2+i\sqrt{3}/2.$