Multiplication by Cube Roots

If you've worked through the three methods presented in Multiplication by Square Roots, moving up to cube roots is a snap, with one minor bump in the road. Why not take a moment to review the techniques presented there. Then try out this problem:

Exercise 1.1: Find the product of 3.2 and the cube root of 27.5.

Let's skip ahead to Method #2. (We'll come back to the first approach later when you'll see why I've postponed it for the nonce).

Method #2: Recall that with this tack, we use commutativity of multiplication to reorder the operands, and then proceed with "ordinary" multiplication, i.e., the multiplication you probably learned first. Here are the steps:
  1. Set the cursor at 27.5 on the K scale. Since this has a two-digit whole number part, you'll use the second 27.5, of course.
  2. Align the left index with the hairline.
  3. Move the cursor to 3.2 on the C scale.
  4. Read the result under the hairline on the D scale: 9.66.
Pretty straightforward, but obviously from time to time you'll have to make an index-swap if the second operand is off-scale. Anyway, it worked out nicely here, and this is what it looks like. Remember, click the photo to enlarge it so you can see the details.

Left index on K:27.5, cursor on C:3.2, result at D:9.66

Method #3: Here we are using the "divide by a reciprocal" technique, which you'll recall is guaranteed to keep one of the indices on-scale. Follow these steps:
  1. Set the cursor at the proper 27.5, as usual.
  2. Align 3.2 on the CI scale with the hairline.
  3. Move the cursor to the right index.
  4. Read the result under the hairline on the D scale: 9.66.
This is really fast and easy on a Rietz style slide rule. With the Pickett 1006-ES I'm using here, a duplex flip is required midstream, hence the three pictures. (That's because CI is on the reverse side of the slide. I suppose if you were going to do a bazillion similar products with cube roots, you could always reverse the slide. But with a duplex rule, it really isn't that big of a deal to simply turn the entire unit over). Anyway, it looks like this:

Left index on K:27.5, then duplex-flip
Align CI:3.2 with the hairline
Cursor to right index, result at D:9.66

Either of the two above methods will be available on most slide rules, which is why I introduced them first. However, the remaining approach (which mimics Method #1 for multiplying by square roots) uses a scale not often found. Indeed, out of my collection only one, the Faber-Castell 2/82, sports the so-called K' scale. It's analogous to the B scale, but for cube roots. For clarity, B is to A (for square roots) as K' is to K (for cube roots).

Method #1 (finally):
  1. Set the cursor at 3.2 on D scale.
  2. Align the left index with the hairline.
  3. Move the cursor to 27.5 on the K' scale.
  4. Read the result under the hairline on the D scale: 9.66.
Because of the way this Faber-Castell is laid out, a duplex-flip is required:

Cursor on D:3.2
Align left index hairline, duplex-flip, then cursor to K':27.5
Duplex-flip, then result at D:9.66

Wrap-Up: All things considered, that was a lot of contortions, just to avoid rearranging the order of the operands. For that reason, surely Method #3, above, is the simplest, just remembering to put the cube root first.

Anyway, it seems to me that the K' scale would be far handier if it appeared on the same side with the C and D scales, tagging along with A and B.

Next installment: Multiplication by Cubes