Cube Root Calculation, Explained

Date: 04/18/2002 at 10:22:01
From: Harsha Ppriya Subramanya
Subject: Cube root manually

Dear Dr. Math,

It was good to see the way you outlined to calculate the cube root 
manually, but I wasn't able to understand. I would appreciate it if 
you could show the same for a small number like 8 (whose cube root we 
know to be 2) and for a number like 9. This would help me and any 
student to understand the method clearly.

Regards,
Harsha Priya Subramanya

Date: 04/18/2002 at 12:56:56
From: Doctor Peterson
Subject: Re: Cube root manually

Hi, Harsha.

Which method did you see and not understand? We describe several 
methods in the links here:

   Square/Cube Roots Without a Calculator - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html#cube 

The method shown in

   Cube roots - Dr. Math archives
   http://mathforum.org/library/drmath/view/52606.html 

uses 8 as an example. This is the easiest method to use and remember. 
If you want the other method demonstrated again, let me know.    

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/18/2002 at 14:08:44
From: Harsha Ppriya Subramanya
Subject: Cube root manually

Dear Dr. Math,

This is the link that I need some explanation on. Probably if you 
could illustrate the same for a number like 8 or 9 it would be 
helpful.

   Cube Root by Hand
   http://mathforum.org/library/drmath/view/52605.html 

This method looks good only if it could be understood.

Regards,
Harsha Priya Subramanya

Date: 04/18/2002 at 14:45:13
From: Doctor Peterson
Subject: Re: Cube root manually

Hi, Harsha.

As I suggested, this is a more complex method, hard to remember and 
requiring a lot of guessing, so the other method is probably 
preferred, but it does have its advantages where humans have to do all 
the work, simply because you never have to divide by a long decimal. 
Let's take the cube root of 8:

    3 _2.________
    \/ 8.000 000
       8
       -
       0

The first step is to "Find the largest cube of a single-digit integer 
less than it"; this is 2^3=8, so we write 2 as the answer, and 
subtract 8, giving a remainder of 0. We're done!

Okay, that was too easy. Let's try 9 this time:

    3 _2.________
    \/ 9.000 000
       8
       -
       1 000

I've brought down the next group of three digits, and we have to find 
the next digit in the answer. This will be the digit x for which

    (300a^2 + 30ax + x^2)x

is less than 1000. (I'm calling the answer so far, 2, by the name 
"a".) To do that, we note that 300a^2 = 300*2^2 = 1200, and 
30a = 30*2 = 60. We then write

    (1200+60*_+_^2)*_=1000

Well, since 1200 is already too big to divide 1000, this digit has to 
be 0. We continue:

    3 _2.__0_____
    \/ 9.000 000
       8
       -
       1 000
           0
       -----
       1 000 000

This time a = 20 (we ignore the decimal point), so 300a^2 = 120000 and 
30a = 600, so we write

    (120,000+600*_+_^2)_ = 1,000,000

For a first guess, as suggested at the bottom of the page, we can 
just divide 1,000,000 by 120,000, which gives x = 8; then

    (120,000+600*8+8^2)8 = 124,864*8 = 998,912

which does the job quite nicely. So now we have

    3 _2.__0___8_
    \/ 9.000 000
       8
       -
       1 000
           0
       -----
       1 000 000
         998 912
       ---------
           1 088

I'll let you continue if you wish; the correct cube root is about 
2.080083823.

For comparison, the other method works like this:

    First guess:  2

    Second guess: 2/3 * 2 + 1/3 * 9/2^2 = 1.33333 + 0.75 =
                  2.0833

    Third guess:  2/3 * 2.0833 + 1/3 * 9/2.0833^2 =
                  1.38886 + 0.69122 = 2.08008

We've already got five significant digits correct. The other method, 
of course, guarantees that every digit we've calculated is correct, 
and probably took less actual calculation if you take into account 
every big square and division I had to do this way; so it's not too 
bad, if you can remember it all.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/