Ray's New Higher Arithmetic · Arts. 366–370 · Unit 25: Involution
108. Powers
Goal: Your student raises whole numbers, fractions and decimals to powers, and uses exponents correctly.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Chapter XIX, Involution: power, root, exponent, square and cube, the rule, notes on multiplying powers and raising a power to a power, powers of fractions and decimals, and examples to show by involution (answers printed).
The lesson, step by step
Warm-up
Ask: What is 3 × 3? 3 × 3 × 3? 2 × 2 × 2 × 2 × 2?
The idea
Read:
A power of a quantity is either that quantity itself, or the product of a number of factors each equal to that quantity.
The root of a power is one of the equal factors which produce the power.
The second power of any number is called the square, because the area of a square is numerically obtained by forming a second power.
The rule
Read:
Multiply the number by itself, and continue the multiplication till that number has been used as factor as many times as are indicated by the exponent.
and the shortcut:
the product of any two or more powers of a number is that power whose degree is equal to the sum of their degrees.
Careful with exponents
Show the book's warning: 2³ × 2² = 2⁵, but (2³)² = 2⁶. Have your student work both out (32 and 64) to see they differ.
Fractions and decimals
Read:
Any power of a fraction is equal to that power of the numerator divided by that power of the denominator.
The square of a decimal must contain twice, and its cube, three times as many decimal places as the root
Practice
Read the instruction "Show by involution, that:" Your student works the list, for example 14³ = 2744, 6⁵ = 7776, (.02)³ = .000008. The answers are printed in the problems; the work is to prove them.
Wrap-up
Ask: Is (½)² bigger or smaller than ½? Why?