2
Zero and negative exponents, and the quotient law
Mathematics Prep 2 Term 1, Unit 1, Lesson 5
GoalUse a⁰ = 1, a⁻ⁿ = 1/aⁿ, the quotient law and (a/b)ⁿ.
New words
zero exponent · a⁰ = 1 for any a ≠ 0negative exponent · a⁻ⁿ = 1/aⁿ: it means ‘one over’, not ‘negative’quotient law · aᵐ ÷ aⁿ = aᵐ⁻ⁿreciprocal · (a/b)⁻ⁿ = (b/a)ⁿ
Explanation
Watch the pattern of powers of 2: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each time the exponent drops by 1 we divide by 2. Keep going: 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8. So a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ. The quotient law follows: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, because 5⁷ ÷ 5⁴ = 5³ and 3² ÷ 3⁵ = 3⁻³ = 1/27. For fractions, (a/b)ⁿ = aⁿ/bⁿ and (a/b)⁻ⁿ = (b/a)ⁿ. Powers of 10 give the metric prefixes: 10³ = 1000, 10⁻² = 0.01 and 0.001 = 10⁻³. Careful: 2⁻³ = 1/8, not −8, and 0⁰ is not defined. In a quotient we subtract the exponents; we do not divide them.
| Power | Value |
|---|---|
| 2⁴ | 16 |
| 2³ | 8 |
| 2² | 4 |
| 2¹ | 2 |
| 2⁰ | 1 |
| 2⁻¹ | 1/2 |
| 2⁻² | 1/4 |
| 2⁻³ | 1/8 |
Worked examples
Simplify 2⁻³ × 2⁵. Same base, so add the exponents: 2⁻³⁺⁵ = 2² = 4. Check: 1/8 × 32 = 4.
Find 3² ÷ 3⁵. Subtract the exponents: 3² − 5 = 3⁻³ = 1/3³ = 1/27. And (2/3)⁻² = (3/2)² = 9/4.
Powers of 10. 3 km = 3 × 10³ m = 3000 m. 5 mm = 5 × 10⁻³ m = 0.005 m. And 0.0001 = 10⁻⁴.
Class activity
Pattern game. Write the powers of 3 from 3⁴ down to 3⁻³ in a column. Divide by 3 each time. Do the same with 10, from 10³ down to 10⁻³. What happens to the number of zeros in the decimals?
Practice
1
Find 7⁰ + 4⁰.
1 + 1 = 2.
2
Write 4⁻² as a fraction.
1/4² = 1/16.
3
Find 6⁵ ÷ 6³.
6² = 36.
4
Find (2/5)⁻².
(5/2)² = 25/4.
5
Write 0.0001 as a power of 10.
10⁻⁴.