Multiplication by Cubes

By this point you should be well skilled at multiplying by squares, square roots and cube roots. And index-swapping or duplex-flipping should seem pretty natural, too. So, bumping things up a notch to products with cubes ought to be trivial. But, if you need to, review the previous entries in this series of Special Products techniques to get up to speed.

Just like when multiplying by squares, there are two methods for multiplying by cubes. Let's try a problem with each.

Exercise 2.1: Find the product of 4.7 and 7.5^3.

Method #1: Here we'll use the old (original) method of multiplication, recognizing that an index-swap might be needed if the set-up takes you off-scale for the second operand.
  1. Set the cursor at 4.7 on the K scale. Any one of the three copies of 4.7 will do just fine in this simple problem.
  2. Align the right index with the hairline. (Did you see the index-swap coming?)
  3. Move the cursor to 7.5 on the C scale.
  4. Read the result under the hairline on the K scale: 1.98 (1980 after adjusting for order of magnitude).
Again for review, the computation actually occurs on the K scale, but by measuring 7.5 on the C scale (which is three times longer), we are in effect multiplying by the cube of 7.5. Here's how it appears:

Right index at K:4.7, cursor on C:7.5, result at K:1.98

Method #2: This is the old reliable business of dividing by a reciprocal, which really ought to be SOP in general.
  1. Set the cursor at 4.7 on the K scale.
  2. Align 7.5 on the CI scale with the hairline.
  3. Move the cursor to the left index.
  4. Read the result under the hairline on the K scale at 1.98.
I'll remind you again that on my Pickett 1006-ES rule, shown here, CI appears on the reverse side of K, implying some duplex-flipping. That won't be the case if you're using a common Rietz type rule. 

Cursor at K:4.7, then duplex-flip
Align CI:7.5 with hairline, then duplex-flip
Cursor to left index, result at K:1.98

Wrap-Up: I suppose, unless you have some specific aim in mind within a chained operation, Method #2 using the CI scale ought to be your general purpose go-to technique. The absence of index-swapping makes it quite appealing, and somehow working directly with the two operands seems more direct, rather than using the left or right index as an intermediary.

Next installment: Multiplication by π and π/4