Ray's New Higher Arithmetic · Art. 378 · Unit 26: Evolution

112. Extracting the cube root

Goal: Your student understands the cube-root method from a real cube and extracts cube roots by the book's rule.

⏱ About 45 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil
  • Sugar cubes or small blocks (optional)

What's in the book

The first explanation of cube root (a cube of 13824 cu. in.), remarks, a second explanation by the cube of tens and units (74088), the rule, notes, and examples (answers printed).

The lesson, step by step

  1. Warm-up

    Ask: 2³? 20³? 30³? Is 13824 between 20³ and 30³? (Yes: 8000 and 27000.)

  2. Work the book's example

    What is the edge of a cube whose solid inches number 13824?

    The edge is between 20 and 30 inches. Take away the 20-inch cube (8000): 5824 cu. in. remain. They make three flat slabs, three long bars and a little cube, all of the same thickness. The slabs' faces total 3 × 400 = 1200 sq. in., so try a thickness of 4. The book adds them up: 1200 + 80 × 3 + 16 = 1456 sq. in., and 1456 × 4 = 5824. The edge is 24 inches.

  3. Build it

    If you have blocks, build a 2 by 2 by 2 cube, then grow it to 3 by 3 by 3. Point out the three slabs, three bars and corner cube that are added.

  4. The trial divisor

    Read the remark:

    Three times the square of the tens is the convenient trial divisor.

    Work the second explanation, √³74088: tens 4 (64000), remainder 10088, trial divisor 4800. Try 2: 4800 + 240 + 4 = 5044; × 2 = 10088. The root is 42.

  5. The rule

    Read the start of the rule:

    Beginning with units, separate the number into periods of three figures each;

    Then follow the steps in the book with ∛19683 together.

  6. Practice

    Your student works ∛512, ∛7301384 and a few more. Read the note:

    Should any product exceed the dividend, the quotient figure is too large.

  7. Check and wrap-up

    Cube each answer to check. Ask: How many figures does the cube root of a 7-figure number have? (3: one for each period of three.)

Afterwards

Cube root is slow. One or two problems a day for several days is better than many at once.

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