Cube Roots and Cubes

If you've licked the previous installment covering square roots and squares using the A and B scales, finding cube roots and cubes will be a cinch. You see, the K scale for them is built analogously. In this case, it's divided into thirds, corresponding to one-third of a logarithm.

In a nutshell, the triplicates on the K scale are each one-third the length of what appears on the C or D scales. Recalling that log(x1/3) = 1/3 * log(x), we've then got a working method at hand.

So when finding cube roots, you'll need to locate the argument on the K scale and in the correct segment. The first third is for arguments from 1.0 to 10.0, the second for 10.0 to 100.0, and the last for 100.0 to 1000.0. Clearly this is cyclical and can be extended for larger numbers. Thus, arguments of one, two and three whole number places continues on to four, five and six, and so on. All because of the pleasant nature of the decimal number system.

I think you know what's coming, so let's jump right in with some problems.

Cube roots


Exercise 1.1: Find the cube root of 3.8.

Since the whole number part of the argument consists of a single digit, we know we should locate it on the first third of the K scale.
  1. Set the cursor to 3.8 on the K scale.
  2. Read the result under the hairline on the D scale: 1.56.
Here's how it appears on my favorite pocket rule, the Pickett 1006-ES. You can click the photo to magnify it:

Cursor to K:3.8, result at D:1.56

Exercise 1.2: Find the cube root of 38.

Here the argument lies between 10.0 and 100.0, so we instantly know we'll be using the second span on the K scale.
  1. Set the cursor to 38 (3.8 in the second span) of the K scale.
  2. Read the result under the hairline on the D scale: 3.36.
In summary:

Cursor to K:38, result at D:3.36

And finally, we should see a problem that takes us to the last third of the K scale.

Exercise 1.3: Find the cube root of 380.
  1. Set the cursor to 380 (3.8 in the third span) of the K scale.
  2. Read the result under the hairline on the D scale: 7.24.
Pictorially we have:

Cursor to K:380, result at D:7.24
As mentioned above, we could continue onward to the arguments 3800, 38000, 380000, and so on. Just keep cycling around the three portions on the K scale and remember (trivially) that the cube root of the multiplier 1000 is 10, the cube root of 10000 is 100, etc.

Cubes


As you'd expect, computing cubes is just a matter of working from the D scale back to the K scale. This is not all that surprising, since measuring the argument on D gives a length three times longer than how it would appear on K. Let's try a problem.

Exercise 2.1: Compute 453.

Before beginning, recall that the D scale runs from 1.0 to 10.0. This argument is in the next order of magnitude, so think of it as 4.5 * 101. Hence, we expect to tag on three zeros after cubing (1000 = 103) to whatever 4.53 is.
  1. Set the cursor to 45 (4.5) on the D scale.
  2. Read the number under the hairline on the K scale: 92.
  3. Tack on three zeros to arrive at the result: 92,000.
It looks like this:

Cursor to D:45, result at K:92,000

With just a little practice, you'll soon become accustomed to the three sections of the K scale: one digit, two digits, three digits, then recirculating to four digits, five digits, six digits, on so forth. Likewise, when working backwards to find cubes: times one, times ten, times one-hundred, times one-thousand, etc.

Work a few more problems of your own devising just to make sure you've got the drift of computing cube roots and cubes using the K and D scales. You'll find additional info in the sections Multiplication by Cube Roots and Multiplication by Cubes.

Next installment: Quartics