Step 13: Cubes and cube roots

Picture of Cubes and cube roots
Notice the K scale at the very bottom of the virtual slide rule. It is three copies of the C and D scales made smaller and laid end to end. It can be used for finding the cube or the cube root of any number. 

Finding the cube of a number is very much like finding the square of a number. Find the number on the D scale. Find its cube on the K scale. See the first graphic. By now you can recognize the C and D scales, even though they are not labeled. The hairline is over the 3 on the C and on the D scales. With you eye follow the hairline downward to the K scale, which is the very bottom scale. Notice that the hairline crosses over the number 27 on the K scale. The cube of 3 is 27. Look at the 4 on the C and D scales. If you look down to the K scale, you can see 64 is directly below it. The cube of 4 is 64. Look at the 5 on the C and D scales. Look down to the K scale. The number below it is 125. The cube of 5 is 125. You can do the same with 6 on the C and D scales. Below (if you could read it well) is 216. The cube of 6 is 216. (Although you cannot accurately read the 6 in 216 on the K scale, you know that 6 x 6 = 36. The last digit in 36 is 6. Again 6 x 6 results in a number that ends in 6. So, even though you cannot read it accurately from the slide rule, you know the last digit in the cube of 6 is also a 6. You can accurately read 21. Add the 6 you know must be part of the number and you have 216. This is a way you can frequently determine a digit that is beyond what you can read accurately on the slide rule.)

Finding the cube root of a number is more complicated than finding the cube of a number. See the second graphic. Basically, you mark off the digits in any number into clusters of three beginning at the decimal point. So, 1,200 would be marked off as 1 + 200. After groupings of three have been marked off, pay attention only to what is remaining. If there is one digit remaining, use the far left third of the K scale. The setup on the slide rule for finding the cube root of 1,200 is shown in the first graphic. Notice the hairline is set at 1,200 on the K scale. Read the answer on the D scale. The numbers on the D scale are 106 plus a tiny bit more. An electronic scientific calculator indicates that the cube root of 1,200 is 10.627.

Find the cube root of 12,000. The setup would be the same, except that the middle of the three sections in the K scale would be used. This is because two digits are left after removing groups of three digits. The digits one can read on the D scale are 229. The electronic scientific calculator indicates the cube root of 12,000 is 22.894.

Find the cube root of 120,000. After removing the first cluster of three digits, three digits remain. Use the far right segment of the K scale. The numbers indicated on the D scale are 494. Checking an electronic scientific calculator, the exact cube root of 120,000 is 49.324. 

There are rules for placing the decimal point when calculating cube roots. They are somewhat involved. In the last step I will link a couple of manuals so that those who wish to become very proficient at cubes and cube roots can learn the exact rules. Or, you can do some guessing in your head and know where to place the decimal point. For example, when calculating the cube root of 12,000 above you know the significant digits are 229. You could guess the number is about 20 just for a test. 20 x 20 is 400. 20 x 400 is 8,000. That is close enough to 12,000 that you now know where to place the decimal point. 

The process of calculating the cube root of a number smaller than 1 also has its special rules. Rather than bog down this Instructable with them, I would refer you to a manual I will link in the last step.