Four Methods for Extracting Cube Roots

Date: 03/03/98 at 08:30:04
From: Sadia Rizwan
Subject: Cube and cube root

Find the cube root of 343.

Date: 03/03/98 at 10:40:23
From: Doctor Rob
Subject: Re: Cube and cube root

You didn't say if you know that the answer is a whole number or not.

If it is, you can factor the number by dividing by prime numbers. Once 
you have that, divide all the exponents by 3, and multiply the result 
out.

Example: cube root of 1728. Factor

      1728 = 2*864 
           = 2^2*432 
           = 2^3*216 
           = 2^4*108 
           = 2^5*54 
           = 2^6*27 
           = 2^6*3*9 
           = 2^6*3^3.  

Since 6/3 = 2 and 3/3 = 1, the cube root is 2^2*3 = 12.

If the answer is not a whole number (or even if it is), there are 
three other ways I can think of to find a cube root:

  1. Use a calculator. Probably you would not be asking this question
     if this were an option.

  2. Guess and check. Try cubing 5: 5^3 = 125, too small, so the
     answer is greater than 5. Try 10^3 = 1000, too big. Answer is
     less than 10. Try 8^3 = 512, too big. Answer is < 8. Try 
     6^3 = 216, too small. Answer is > 6. Continue narrowing the range
     in which the number lies until you find the answer as accurately
     as you need. Most efficient is to pick your new trial number
     exactly in the middle of the remaining range of possibilities.

  3. Iteration. Start with a guess, like x(0) = 5. For each n >= 0, do

          x(n+1) = [2*x(n)^3 + 343]/[3*x(n)^2].

     Do this until one of your x(n)'s is very close to the one 
     before it. If x(0) = 5, then

          x(1) = (250+343)/75 = 7.9067,
          x(2) = 7.1000,
          x(3) = 7.0014,
          x(4) = 7.0000,

     and all further x(n)'s will equal 7.0000, to four decimal places.

     This is how any calculator with a cube-root key works.

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