How to do cube roots in your head March 13, 2011Posted by Ezra Resnick in Math.
The next time you’re at a math-friendly party (or anywhere you don’t mind not being invited back to), ask one of the people nearby to secretly choose any two-digit number, cube it (he or she may use a calculator), and tell you the result. Watch your audience’s jaws drop and their eyes fill with admiration as you immediately name the original number! The cube root of 10,648? That’s 22, of course. How about 658,503? Why, it’s 87. I can go all night.
I was taught this trick by a friend of my uncle’s, the kind of person who always has a marked deck of cards and a fake thumb in his pocket. Five minutes of practice should be enough to master it — all you need to remember are the cubes of the numbers 0 through 9:
03 = 0
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
You can deduce the tens digit of your mystery root by knocking the last three digits off of the given cube, and seeing where the remainder falls among the values above. For instance, if you’re left with something smaller than 8 (i.e., the cube is less than 8,000), then the tens digit of your mystery root is 1. If you’re left with something greater than 7 but less than 27, then the tens digit you want is 2. And so on — if you’re left with 729 or more then the tens digit you’re looking for is 9.
What about the units digit? That’s even easier: it can be deduced directly from the units digit of the given cube. Notice that the units digits of the cube values above are all different — in most cases, it’s the same as the root number itself! The four exceptions pair up nicely: 2 and 8 map to each other, as do 3 and 7.
So to put it all together: say the person you want to impress gives you the cube value 175,616. Since 175 is between 125 and 216, the first digit you’re looking for is 5; and since the cube value ends with 6, so must its root — 56 it is.
I think my work here is done.