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### 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/
```
Associated Topics:
High School Square & Cube Roots

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