Mental Math Tricks: Finding Cube Roots of Large Numbers

Date: 06/07/2004 at 11:44:34
From: Karim
Subject: finding the cube root of very large numbers in your head

A friend asked me to pick a number between 100 and 200, cube it, and
give him the answer.  After thinking about it, he gave me the original
number that I had cubed--the cube root of the number I gave him.  How
does he do this in his head without a calculator?

Date: 06/07/2004 at 13:07:13
From: Doctor Douglas
Subject: Re: finding the cube root of very large numbers in your head

Hi Karim.

Thanks for writing to the Math Forum.  This is an impressive trick!
Your friend first needs to have memorized the cubes of the numbers
from 0 to 9 (or 1 to 10):

  0 x 0 x 0 =  0        5 x 5 x 5 = 125
  1 x 1 x 1 =  1        6 x 6 x 6 = 216
  2 x 2 x 2 =  8        7 x 7 x 7 = 343
  3 x 3 x 3 = 27        8 x 8 x 8 = 512
  4 x 4 x 4 = 64        9 x 9 x 9 = 729

Now, the key idea is that the units digit of these ten numbers are all 
different, and they are easy to memorize, 0,1,4,5,6, and 9 all have 
cubes that end in the same number, and the remaining four numbers come 
in pairs (2,8) and (3,7):  e.g. 8 x 8 x 8 ends in 2 and 3 x 3 x 3 ends 
in 7.

So if you have a large number that is the cube of a whole number, such 
as 4657463, you simply have to check the last number to determine what 
the units digit of the original number is.  Since this large number 
ends in 3, its cube root must end in 7.

Now, this takes care of extracting the units digit.  The other part of 
the trick requires your friend to memorize the next ten cubes:

  10 x 10 x 10 = 1000     15 x 15 x 15 = 3375
  11 x 11 x 11 = 1331     16 x 16 x 16 = 4096
  12 x 12 x 12 = 1728     17 x 17 x 17 = 4913
  13 x 13 x 13 = 2197     18 x 18 x 18 = 5832
  14 x 14 x 14 = 2744     19 x 19 x 19 = 6859

By comparing the leading four digits of your number with these, your 
friend can estimate what must be the leading two digits of your cube 
root.  In the example of 4657463, the leading four digits are 4657, 
and this falls between 16 x 16 x 16 = 4096 and 17 x 17 x 17 = 4913.  
Thus the original cube root is (by incorporating our units digit 
information from above) 160 + 7 = 167:  4657463 = 167 x 167 x 167, 
obtained with essentially no arithmetic computation whatever!

Note that there cannot be any ambiguity in the range of these leading
four digits, since the units digit will add at most 9^3 = 729, a 
number less than 1000.  In other words, whatever the units digit is,
its cube will never carry over into the thousands place when the 
number is cubed.

Note that your friend does not have to restrict your number to between
100 and 200.  Cubes that are less than 1,000,000 (100^3) can be easily
handled since your friend has already memorized the cubes of the first
ten numbers.  For example, try 314,432:

  314432/1000 is approximately 314, which is between 6 x 6 x 6 and
  7 x 7 x 7, so the original number must have been 60 + something.
  Since the given cube ends in 2, the number must have been 68.

I hope that explains how the trick works.  By memorizing just twenty
numbers you will also be able to perform this trick.  And with some
practice, I think you'll be able to extract the cube roots of perfect
cubes of numbers between 0 and 200 in about 10 or 15 seconds!

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

Date: 06/07/2004 at 13:53:28
From: Karim
Subject: Thank you (finding the cube root of very large numbers in
your head)

Thank you very much...you're a star!