So, what does that give us?

Well, what is the square root of negative 1 squared?

Even if we don't know what the square root of -1 is, if we square whatever it is,

the square root and

the square should cancel each other out, so this would really be- 1.

That's the idea of the square root of -1, and you square it get- 1 back.

And what is the square root of -1 cubed?

Well that's really the square root of -1 squared x another square root of -1.

But we just figured out the square root of -1 squared is -1, and

then there's this extra square root of -1.

And so all together let's simplify 2 cubed is = 8.

Then we have 3 x 2 squared, that's 12 x the square root of -1.

And then we have 3 x 2 x -1.

Against 6 x -1 is- 6.

And here we have- 1 x square root of -1.

Now we can combine at least like terms.

We can combine this 8 and this negative 6, 8- 6 is = to 2.

And we have 12 x whatever square root of -1 might be, and

we're subtracting from that 1 x the square root of -1.

So all together we have 11 x the square root of -1.

What is 11 x the square root of -1?

Well if square roots kind of interact normally, even with negative numbers,

then we might be able to pull that 11 inside the square root.

And that becomes 121 on the inside, and so

it should be then be able to 2 + the square root of- 121.

And that's what we wanted to show.

And so Bombelli was indeed correct,

if I take the cubed root of this right hand side,

the cubed root of 2 + the square root of -121.

Is indeed equal to the [INAUDIBLE] of this left hand

side 2 plus the square root of -1.

And then similarly when you replace this + right

here with a- then this plus also becomes a -.

And then we add those two terms up, you end up with that 4,

which was the solution to the cubic equation we were looking at.

And so all of a sudden solving a perfectly real problem,

that cubic equation Bombelli was looking at required accepting that

complex numbers are important objects, and that we can do calculations with them as

if they were objects that behave according to the rules of our real numbers.

This is considered the Birth of Complex Analysis.

It showed that perfectly real problems require complex arithmetic for

their solutions and

it also showed that we need to be able to manipulate complex numbers according

to the same rules we're used to from real numbers, like the distributive law.

And we need to be able to add them, we need to find out how to multiply them.

So we need to study the rules of doing arithmetic with complex numbers.

And that's what we'll study next.