Melody

Thanks Chris :)

Guest, you are certainly correct. !

There is only 1 real roots and since most students only learn about real numbers you are 100% correct.

BUT

If you study maths for long enough you will learn about  imaginary. numbers.

Two of these roots are complex roots and that means that they are partly imaginary.

It is all based on the idea that    \(\sqrt{-1}=i\)

That is the basic element of the whole imaginary number system.

Lets see how this works,

I have said that one of the cubic roots is     \(-1-\sqrt3\;i\)

so if this is true then

\(2^3=(-1-\sqrt3\;i)^3\)

\(LHS=8\\ RHS=\;  (-1-\sqrt{3}\;i)(-1-\sqrt{3}\;i)(-1-\sqrt{3}\;i)\\ RHS=\; - \;\;\;\;(1+\sqrt{3}\;i)(1+\sqrt{3}\;i)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS=\; -\;\;\;\; (1+2\sqrt3\;i+3*-1)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS=\; -\;\;\;\; (1+2\sqrt3\;i-3)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS= \;-\;\;\;\; (-2+2\sqrt3\;i)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS= \;-\;\;\;\;-2 (1-\sqrt3\;i)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS= \;\;\;\;\;2\;\; (1-\sqrt3\;i)\;\;\;\;(1+\sqrt{3}\;i)\\ RHS= \;\;\;\;\;2\;\; (1^2-(\sqrt3)^2(i^2))\\ RHS= \;\;\;\;\;2\;\; (1^2-(3)(-1))\\ RHS= \;\;\;\;\;2\;\; (1+3)\\ RHS=\;\;\;\;\;\ 8\\ RHS=LHS\\ \mbox{therefore one cubic root of 8 is }\:\: (-1-\sqrt{3}\;i) \)

Ther are 3 roots.

The first one is easy to find it is 2

There are 2pi radians in a circle and we want 3 roots so each will be 2pi/3 radians apart.

The roots will be

2,     2 cis(2pi/3)  and    2cis(-2pi/3)

\(cos(\frac{2\pi}{3}) = - cos (\frac{\pi}{3}) =\frac{ -1}{2}\\ sin(\frac{2\pi}{3})= sin(\frac{\pi}{3}) = \frac{\sqrt3}{2}\\ cis(\frac{2\pi}{3}) =\frac{ -1}{2}+i*\frac{\sqrt3}{2}\\ cis(\frac{2\pi}{3}) =\frac{ -1+\sqrt{3}\;i}{2}\\~\\ cos(\frac{-2\pi}{3}) = - cos (\frac{\pi}{3}) =\frac{ -1}{2}\\ sin(\frac{-2\pi}{3})= -sin(\frac{\pi}{3}) = \frac{-\sqrt3}{2}\\ cis(\frac{-2\pi}{3}) =\frac{ -1}{2}-i*\frac{\sqrt3}{2}\\ cis(\frac{-2\pi}{3}) =\frac{ -1-\sqrt{3}\;i}{2}\\ \)

\(\mbox{So the 3 solutions are }\\~\\ \sqrt[3]{8}=2,\qquad and \quad\\ \sqrt[3]{8} = -1+\sqrt{3}\;i\quad and \quad\\ \sqrt[3]{8} = -1-\sqrt{3}\;i, \\\)

I think there may be some sarcasm in your last post Ally.  

You have over reacted to everything.

Lance was not teasing you, he just saw a great opportunity for a bit of word play fun.

His post was HILARIOUS!

Hayley was just showing appreciation of Lance's humour!

Lighten up and enjoy the forum.  Stop thinking that people are out to get you!  

Just enjoy yourself.

We need negative numbers, 

Say our bank balance is $6.

I do not know if the bank has $6 of my money OR if I have $6 of their money (a loan)

But if I say I have +$6 or -$6 then I would know which one I meant.

A better example

The temperature is 10 degrees celcius.  

For me that is very cold but if you live in a very cold country that might be nice weather.

The temperature drops a long way.  It gets way colder.  It gets colder than 0 degrees celcius.

The temperature becomes negative.  If we had no negative numbers we could not talk about temperatures any colder than 0.  That would be really limiting. :)

The same happens when you talk about altitude above and below sea level. :)

Good luck Hayley  :)   

Maybe this will help

An artist wants to make alabaster pyramids using a block of alabaster with a volume of 576 cubic inches. She plans to make each pyramid with a square base area of 3 square inches and a height of 4 inches. At most, how many pyramids can the artist make from the block of alabaster?

The volume of each pyramid

=(1/3)*area of base* height

=(1/3)*    3*3    * 4

=12 cubic inches

576/12 = 48 pyramids

Excellent guest that is another way of presenting it, and it is more elegant than mine.

Tn=(n-1)!

Hi Titanium,

I haven't seen you for a while.

It's nice to see you now:D