Cube roots

Date: 05/30/97 at 09:34:01
From: Elena
Subject: Cube roots

Dear Dr. Math,

How can  I can  figure out the cube root of a number? (i.e., that the 
cube root of 216 is 6).
 
Thank you for helping me!
Elena

Date: 05/31/97 at 08:30:28
From: Doctor Jerry
Subject: Re: Cube roots

Hi Elena,

There isn't a "test" that you can apply to an integer to determine 
whether it is a cube of an integer, other than looking it up in a 
table.

Perhaps you are asking for an algorithm to calculate the cube root 
of any positive number.

If a is the number you want the cube root of, make a guess and 
call it g.  The next guess is

  G = (2/3)*g + (1/3)*(a/(g^2)).

Repeat this until the cube root is found to desired accuracy.

If a = 8, for example, suppose our first guess is 3.  Using the above 
formula, here are the successive guesses:

   g1 = 3
   g2 = 2.2962962963
   g3 = 2.03658740252
   g4 = 2.00065335855
   g5 = 2.00000021335
   g6 = 2.00000000000 ( to the accuracy of my calculator)

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