Mental Math Tricks: Finding Cube Roots of Large NumbersDate: 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! |
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