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Using Mental Arithmetic to determine:
Cube Roots

Immediately know the cube-root of of any whole cube from 1 to 1,000,000 

The Trick:   To determine the cube-root of any whole cube, from one to one million, within a second or two, without any calculator...

1 x 1 x 1 =   1   (cube-root of  1 = 1)
2 x 2 x 2 =   8   (cube-root of  8 = 2)
3 x 3 x 3 =  27  (cube-root of 27 = 3)
4 x 4 x 4 =  64  (cube-root of 64 = 4)
5 x 5 x 5 = 125 (cube-root of 125 = 5)

So what is the cube root of 1,000,000?   That's easy.  It's 100.

But now, how would you instantaneously determine the cube-root of 493,039 ??

That is, how would you get the cube-root of 493,039, within one or two seconds, without a calculator, without paper, and without any writing implement?

Or, for example, what is the cube-root of 912,673?

Answers: 
The cube-root of 493,039 is 79...
The cube-root of 912,673 is 97...
Trust me, you don't have to memorize the cube-roots...

Instead, i will teach you how to do this in your head, instantaneously...

All you have to do is memorize the primary cubes from one to ten:
1 x 1 x 1 =   1   (cube-root of 1 = 1)
2 x 2 x 2 =   8   (cube-root of 8 = 2)
3 x 3 x 3 =  27  (cube-root of 27 = 3)
4 x 4 x 4 =  64  (cube-root of 64 = 4)
5 x 5 x 5 = 125 (cube-root of 125 = 5)
6 x 6 x 6 = 216 (cube-root of 216 = 6)
7 x 7 x 7 = 343 (cube-root of 343 = 7)
8 x 8 x 8 = 512 (cube-root of 512 = 8)
9 x 9 x 9 = 729 (cube-root of 729 = 9)
10 x 10 x 10 = 1,000 (cube-root of 1,000 = 10)...

Now here's how you do it: 

First, give a friend a calculator, and ask them to choose any number from 1 to 100, and to use the calculator to cube it... 

In one of the above examples, the number 79 was chosen.   Cubing it, using a calculator, 79 x 79 x 79 = 493,039...

The person holding the calculator tells you the cube, and your job is to get the cube-root immediately, using "mental arithmetic."

The way you do it is this:

If the large number ends in 1, 9, or 4, 5, or 6... then the cube-root also ends in 1, 9, 4, 5, or 6...   If the large number ends in 2 or 3, the cube ends in 8 or 7...  If the large number ends in 7 or 8, the cube-root ends in 3 or 2... (see above list of primary cubes)

Let's look at the example above:   79 x 79 x 79 = 493,039

So if i tell you the cube is 493,039, you know immediately that the cube-root ends with a 9...

But how do you determine the rest of the cube-root??...

Since we're only dealing with cubes of numbers from 1 to 99 (while the cube-root of 1,000,000 is 100), we know that the cube-root will be either a 1 or 2-digit number...   Anything between 1,000 (the cube of 10) and 1,000,000 (the cube of 100) will be a 2-digit number.

Now, look to the left of the comma in the larger number 493,039.   The number is 493...  Go back to the list of primary cubes from 1 to 10, and 493 will fit between 7 and 8...  The cube-root of 493,039 is thus 79...

Try another: 262,144...  The cube ends with a 4, thus the cube-root also ends with a 4...  The thousands is 262, which is between 216 (the cube of 6) and 343 (the cube of 7)...   Thus, the answer is 64...

With a little practice, it gets very easy...  You'll amaze your friends...

All you have to do is memorize the primary cubes... and understand this procedure...

See also Rubik's Cube solution (with closed eyes)...

Goodman Livingwell






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