Ray's New Higher Arithmetic · Arts. 379–382 · Unit 26: Evolution
113. Cube roots of products and fractions; any root
Goal: Your student takes cube roots of products and fractions, and finds higher roots by taking square and cube roots in turn.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
The cube root of a product and of a fraction, remarks, practice examples (answers printed), then extraction of any root by factoring the index.
The lesson, step by step
Warm-up
Ask: What is ∛8? ∛27? ∛216?
Products
Read:
The cube root of any product is equal to the product of the cube roots of the factors.
Show the book's example: ∛250 × ∛4 × ∛648 × ∛9 = ∛(125 × 8 × 216 × 27) = 5 × 2 × 6 × 3 = 180.
Fractions
Read:
The cube root of a common fraction is equal to the quotient of the cube root of the numerator by the cube root of the denominator.
If the denominator is not a perfect cube, multiply both terms to make it one.
Any root
Read:
the 9th root is the cube root of the cube root, and the 6th root the square root of the cube root.
Try: the 4th root of 65536 is the square root of its square root. √65536 = 256, and √256 = 16.
Practice
Your student works some of the remaining cube-root examples on the page.
Wrap-up
Ask: How could you find a 6th root using what you know? (Take the square root, then the cube root, or the other way round.)