M2C2A: Episode 11, The root of a number?!

A square root of a number y is by definition a number x in which x² equals y. And a cube root of a number y is by definition a number x in which x³ equals y.
But we can also do n roots, in which the nth root of y is a number x in which which xⁿ equals y. But there's a special property about roots:

They may have many answers.

For example, the square root of 4.
2² is 4, but also -2². So can 2 and -2 be the roots of 4?
Yes!
Let's now take a different example:
The cube root of 8.
It may be true, but there are 2 more solutions.
First of all, you'll need to know complex numbers.
A complex number has a real part (2, 4, 7982) and an imaginary part, which can be any number times i.
But what is i?
i is the square root of -1, which doesn't exist, but can be used in many ways.
You can add i (23+i), subtract i (9-i), multiply i (23i) and divide i (i/5).
You can get i to the power of a number (i¹=1, i²=-1, i³=-i, and i⁴=1. This pattern repeats, and iⁿ=i^{n mod 4}).
So the second answer of the cube root of 8 is
-1 + 1.7320508075688772i and the third answer is -1-1.7320508075688772i.
And fourth roots?
Let's say we have the fourth root of 81. There are 4 answers for this:
A) 3
B) -3
C) 3i
D) -3i


We know that 3*3*3*3 is 81, and that -3*-3*-3*-3 is 81, but why 3i and -3i?
3i*3i is -9, because if the square root of -1 is i, then i² should be -1. And because 3*3 is 9, 9*-1 is -9, 3i² is -9. But we want 3i⁴, not 3i². So we square -9 to get 81. Same story with -3i.
So really, the root of a number is much more complex than you might think.
 
-The Roaring Thunder