$($Roots of negative unity: adapted from LM $3\mathrm{.}1-24)$ Let $n$ be a positive integer. Recall

that

$-1=e^i=e^{3i}=e^{5i}=$cdots,

but, if we take the $n$th root, we obtain $n$ distinct values

$e^{\frac{i}{n}},e^{3\frac{i}{n}},e^{5\frac{i}{n}},$dots,$e^{(2n-1)\frac{}{n}}$

due to the periodicity of the complex exponentials; think Euler. These are roots of $x^n+1,$

or $n$th roots of $-1.$

$($a$)$ Write a function at the end of the live script which, given a positive integer $n,$ finds all

$n$ roots of $x^n+1$ at once, using ONE statement. Do NOT use a loop.

Note. The input to the function is a positive integer $n$ and the output of the function

is the $($column$)$ vector of all $n-$th roots of negative unity.

Suggested approach. Begin by writing a script which solves the problem with $n=5.$

First do it using a for$-$loop, e$.$g$.,$ the main fragment would contain

Then think about how to modify it to handle a general value of $n.$ Lastly, how to

think about vectorizing the code so that the loop is replaced by a single statement.

$($b$)$ Run the function with $n=3,5,7,$ and $11.$ For each $n,$ print out all $n$ of these roots

neatly using either disp or fprintf in tabular form. A loop may be used for this part.