Sextics

By this time, having read over the previous sections here on squares, cubes, quartics and quintics, you won't be surprised to learn sextics (sixth powers) are possible with the aid of the K scale. The technique is analogous to what we saw with quartics. In particular:

x⁶ = (x * x)³

We'll carry out the multiplication in parentheses with CI and D (multiplying by a reciprocal), then do a scale-transfer from D to K. Only one slide movement is required.

Exercise 1.1: Compute 1.8⁶.

1. Set the cursor to 1.8 on the D scale.
2. Align 1.8 of the CI scale with the hairline.
3. Move the cursor to the right index.
4. Duplex-flip, if needed.
5. Read the result under the hairline on the K scale: 34.

In the pictorial summary here, you'll note I needed to do a duplex-flip, since the K scale is on the side opposite CI with my Pickett 1006-ES. If you're using a Rietz type slide rule, that won't be necessary.

You can click any pictures here to enlarge them.

Cursor to D:1.8, CI:1.8 to hairline, cursor to right index

Duplex-flip, result at K:34

Observe that the result ended up in the second segment of the K scale, so the result must have a whole number part with an even number of digits. Only 34 makes sense. Or if you prefer, just estimate by thinking of 2⁶ = 64, again a two-digit result.

Exercise 1.2: Compute 22⁶.

1. Set the cursor to 22 (2.2) on the D scale.
2. Align 22 (2.2) of the CI scale with the hairline.
3. Move the cursor to the right index.
4. Duplex-flip, if needed.
5. Read the result under the hairline on the K scale: 113,000,000.

Here's how it goes:

Cursor to D:22, CI:22 to hairline, cursor to right index

Duplex-flip, result at K:113,000,000

To estimate the order of magnitude, I thought of 22 as 2.2 * 10¹. After raising to the sixth power, then, a multiplier of 10⁶ will be needed. Then, estimating with 2⁶ = 64, suggests something with two or more significant digits. Finally, noting that the cursor ended up in the third section of K, we know those digits must make 113 and not 11.3. (That is, the whole number part must have 3, 6, 9, etc. number of digits).

And so there you have it for this sequence of installments: how to compute powers of degree 2, 3, 4, 5 and 6 with just a handful of the commonest scales. Square roots and cube roots also came along for the ride. For anything more exotic, you'll probably want to turn to the log-log scales.

The following installments recap some of this, and then show how to use these powers and roots in various products.

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