Cube Roots of Numbers

Date: 11/05/97 at 12:29:41
From: Jared Martin
Subject: Cube roots of numbers

If x^3 = N, where N is some expression (which could be a constant), 
then you have a degree three equation so there must be three roots. 
If you take i (sqrt(-1)), then the cube root is -i. But since 
x^3 = i is degree three, there should be three different values of x. 

What are they? How do you determine these three values for other 
numbers? Is there a formula? Please help.

Date: 11/05/97 at 16:55:49
From: Doctor Anthony
Subject: Re: Cube roots of numbers

You are quite right that there will be 3 cube roots of a number.  
You do need to work with complex numbers, however, to understand how 
to find the three roots, so if what I show you is not clear, it will 
become so when you have studied complex numbers.

    Suppose  z^3 = 8

Now taking the cube root of each side you would say that z = 2, 
however, there are two other cube roots which we shall now find.

Since cos(2k.pi) = 1 and sin(2k.pi) = 0  where k is any integer,  
we could write the equation

             z^3 = 8(cos(2k.pi) + i.sin(2k.pi))

Take cube root of both sides, and use deMoivre's theorem which shows 
that:

    [cos(x) + i.sin(x)]^(1/3) = cos(x/3) + i.sin(x/3)   to get

            z = 2[cos(2k.pi/3) + i.sin(2k.pi/3)]    k = 0, 1, 2

k=0 gives  z1 = 2(cos(0) + i.sin(0)]  = 2    (the one real root)

k=1 gives  z2 = 2(cos(2.pi/3) + i.sin(2.pi/3)) = 2(-1/2 + i.sqrt(3)/2)

k=2 gives  z3 = 2(cos(4.pi/3) + i.sin(4.pi/3)) = 2(-1/2 - i.sqrt(3)/2)

If we give k more values, 3, 4, 5, ..... we simply repeat the three 
roots already found.

If you represent the three roots on an Argand diagram that has real 
values along the x axis and imaginary values on the y axis, the three 
roots will appear as the three spokes of a wheel, with the z values 
lying on a circle of radius 2 units. One root will lie along the 
positive x axis, and the other two at +120 degrees and -120 degrees to 
the x axis. So the roots are symmetrically spaced round the circle.  
In fact this is always the way that cube roots of a real number will 
look. If you take the cube root of an imaginary number, say i, then 
you still get three spokes but they will be rotated round to lie along 
the 30 degree, 150 degree and 270 degree lines on the unit circle.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/