Powers: 1/3, 1/2, 2/3, 3/2, 2 and 3

This is a preface of sorts and concerns itself with several common roots and powers that'll be needed in the following sections. Before we get to the actual multiplication there, however, it might be wise to investigate the A and K scales first, just to see how squares and cubes (and their inverse functions, roots) are handled. This is really fun, as we discover just what can be accomplished without even touching the slider proper!

A couple of reminders: The mantissas as used by a slide rule (see Some Terminology) are always considered to lie between 1 and 10. Hence, if I say to set the cursor at 18, for instance, you know you'll be moving it to 1.8 and mentally adjusting for the order of magnitude at the end of a computation.

Also, keep in mind that you can always click the photos to enlarge them for closer inspection.

Square roots: Begin by comparing the A and D scales. You'll observe both are logarithmic, but the logs are twice the length on D as they are on A, or if you prefer, those on A are half the length as they are on D. Remembering Napier's rule that (1/2)*log(x) = log(sqrt(x)), we see what's coming.

In short, when you measure out a number on the A scale by aligning the cursor with it, the hairline falls on the square root of that number on the D scale. And obviously, you can work this backwards to determine a square.

Scale A consists of two abutted half-length portions. The first applies to arguments whose whole number part consists of an odd number of digits, e.g., 2, 2.5, 315.2, etc. The second half is for arguments with whole number parts containing an even number of digits, e.g., 20, 25.5, 3100.52, etc.

With that in mind, we're ready to work some problems. We'll begin with square roots. Simply align the cursor with the argument (in the proper half of A, of course) and read the result off on the D scale. The slide is not used.

Exercise 1.1: Find sqrt(5.4).
  1. Align the cursor with 5.4 in the first half of A.
  2. Read the result under the hairline on D: 2.32.
Here's what it looks like on my ever-trusty Pickett 1006-ES:

Cursor on A:5.4, result at D:2.32

Exercise 1.2: Find sqrt(27.8).
  1. Align the cursor with 27.8 in the second half of A.
  2. Read the result under the hairline on D: 5.27.
Cursor on A:27.8, result at D:5.27

Squares: When finding squares, you'll (quite naturally) work backwards from the D scale to the A scale. You can use knowledge of which half you wind up in to help with setting the order of magnitude at the end, if you wish. Otherwise, just estimate by considering an "easy" number close to the argument.

Exercise 2.1: Find 23^2.
  1. Align the cursor with 23 on the D scale.
  2. Read the result under the hairline on A: 5.3.
  3. Mentally adjust the order of magnitude: 530.
Cursor on D:23, result at A:5.3

Exercise 2.2: Find 65^2.
  1. Align the cursor with 65 on the D scale.
  2. Read the result under the hairline on A: 4.23.
  3. Mentally adjust the order of magnitude: 4230.
Cursor on D:65, result at 4.23

Cube roots: This'll be easy now using what you just learned about square roots. Simply switch your attention to the K scale which is made up of three juxtaposed third-length portions. This implies, of course, that you pick the correct portion according to whether the argument's whole number part has 1, 4, 7,... or 2, 5, 8,..., or 3, 6, 9,... digits, i.e., p = (n-1) mod 3 +1, where n is the number of digits, and p which of the three portions to use (left, middle or right).

Exercise 3.1: Find the cube root of 6.6.
  1. Align the cursor with 6.6 in the first third of K.
  2. Read the result under the hairline on D: 1.88
Cursor on K:6.6, result at D:1.88

Exercise 3.2: Find the cube root of 66.
  1. Align the cursor with 66 (6.6) on the middle third of K.
  2. Read the result under the hairline on D: 4.04.
Cursor on K:6.6, result at D:4.04

Exercise 3.3: Find the cube root of 666.
  1. Align the cursor with 666 (6.66) on the right third of K.
  2. Read the result under the hairline on D: 8.7.
Cursor on K:6.66, result at D:8.7

Cubes: You should be getting pretty good at this racket and know what's coming next. To cube a number, we simply start on the D scale and read the result on the K scale, adjusting the order of magnitude as required.

Exercise 4.1: Find 41.5^3.
  1. Align the cursor with 41.5 on the D scale.
  2. Read the result under the hairline on K: 7.2.
  3. Mentally adjust the order of magnitude via estimation: 72,000.
Cursor on D:4.15, result at K:7.2

Exercise 4.2: Find 15^3.
  1. Align the cursor with 15 (1.5) on the D scale.
  2. Read the result under the hairline on K: 3.35
  3. Mentally adjust the order of magnitude via estimation: 3350.
Cursor on D:1.5, result at K:3.35

The Power 2/3: In the above examples we worked the D scale against A, or D against K, yielding squares, square roots, cubes and cube roots. But what if we match up A directly with K? Why then, we have a fast and easy approach to the rational exponents 2/3 and 3/2!

 This is pretty cool, when you think about it for a moment. For example, if we start with an argument on the K scale, then the result on the D scale is a cube root. But continuing from this cube root on D, up to the A scale then squares the intermediate result. We've found x^(2/3) power! And there's really no need to even think about the D scale at all; it's a needless middleman here. Just go directly from K to A, and you're done. Let's try it out.

Exercise 5.1: Find 5^(2/3).
  1. Align the cursor with 5 on the K scale (left one of the trio, of course).
  2. Read the result under the hairline on the A scale: 2.92.
Cursor on K:5, result at A:2.92

Exercise 5.2: Find 12.5^(2/3).
  1. Align the cursor with 12.5 on the K scale (middle portion).
  2. Read the result under the hairline on the A scale: 5.4
Cursor on K:12.5, result at A:5.4
The Power 3/2: It's just as easy to turn things around and work from the A scale to the K scale in order to realize an argument raised to the 3/2 power. Again, there's really no need to think about that interlocutor, D, as playing any role.

Exercise 5.3: Find 22^(3/2).
  1. Align the cursor with 22 on the second half of the A scale.
  2. Read the result under the hairline on the K scale: 10.3.
  3. Mentally adjust the order of magnitude: 103.
Cursor on A:22, result at K:10.3

Exercise 5.4: Find 150^(3/2).
  1. Align the cursor with 15 on the second half of the A scale.
  2. Read the result under the hairline on the K scale: 1.84.
  3. Mentally adjust the order of magnitude: 1840.
Cursor on A:15, result at K:1.84

Coming up later on this Web site will be methods for dealing with arbitrary powers, but you've just seen how to deal with six common ones, and they're typically available with even the simplest of slide rules.

With the prereqs out of the way, let's dig into the heart of the matter: how to multiply by squares and square roots. Click the link below for the next installment.

Next installment: Multiplication by Square Roots