Cube Roots
Jeffkillian.com initially started as a Geocities website when I was in 6th grade. Over the years, it has slowly evolved into a place where I can practice and become familiar with web development as it progresses. I started out coding with Microsoft Frontpage, then moved to Dreamweaver. Initially, there was very little coding, and it was all WYSIWYG. However, as I learned more, I could make it more interactive.


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Cube Roots

A while back somebody introduced me to a website where they taught you how to do cube roots of 6 digit numbers in your head. The only criteria was that the cube root had to be a whole number. For instance, if I was given 250, 047, I could say that the cube root of that was 63 in a matter of seconds. It's a pretty easy formula, and it goes as follows:

i i3 ones digit(k)
0 0 0
1 1 1
2 8 8
3 27 7
4 64 4
5 125 5
6 216 6
7 343 3
8 512 2
9 729 9

All you have to remember is the cubes of the single digits, which you can see in the table. The next step is to realize that each single digit cubed ends in a unique one's digit(0 and 0, 1 and 1, 2 and 8, 3 and 7, etc), and you must memorize that number associated with the cube. To get the initial number that was cubed, j ,you do two steps (one to get the tens digit and one to get the ones digit). To get the ones digit of the initial number, you look at the ones digit(call this k) of the cubed number(j3), and take the j which, when cubed ends in k. To get the tens digit, you simply look at the thousands, tenthousands, and hundred thousands digits of the cubed number. Interpret these as a three digit integer between 0 and 999, and find the number in the second column which is closest to without going over that number. The i that corresponds to that cube is the tens digit of your initial j.

In summary: Get ones digit by looking at the ones digit of the given cubed number. Get tens digit by looking at the floor of the cube root of the thousands, ten-thousands, and hundred-thousands digits (when combined) of the number.

For example, if we are given 148,877. We first look at the ones digit, which is a 7 (bolded). Looking at the table, we go to the third column where 7 is, and move over two columns to obtain 3, which we now know is the ones digit of the initial number. Next, we want to get the tens digit, so we look at the numbers in the thousand, tenthousand, and hundredthousand column, which are 148. Looking at the second column, we see 125, or 5 3, is the highest we can get without going over 148 (216, which is 63) is too high. So that means 57 is our answer, and 573=148,877. I wrote a TI-83 application in high school to do this, but I threw a quick java program together to help if you wanted to practice.

This page has a good explanation of the process, as well as extends it to finding cube roots of 9 digit numbers.


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