Exponents & Powers
Laws of Exponents (Integers)
The laws hold for integer exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ, with a⁰ = 1. So 2³ × 2⁻¹ = 2² = 4.
Example 1: Simplify 2³ × 2⁻¹.
2³⁻¹ = 2² = 4.
Example 2: Simplify (10³)².
10⁶.
Quick recap
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ.
- a⁰ = 1.
✓ Quick check
Simplify 3² × 3³ as a power of 3.
Add the exponents: 3²⁺³ = 3⁵.
What is 5⁰?
Any non-zero number to the power 0 is 1.
Negative Exponents
A negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ. So 2⁻⁴ = 1/16 and 10⁻³ = 1/1000 = 0.001.
Example 1: Find 2⁻⁴.
1/2⁴ = 1/16.
Example 2: Find 10⁻³.
1/1000 = 0.001.
Quick recap
- a⁻ⁿ = 1/aⁿ.
- 10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001.
✓ Quick check
What is 4⁻¹?
4⁻¹ = 1/4 = ¼.
What is 2⁻³?
2⁻³ = 1/2³ = 1/8.
Standard Form & Comparing Numbers
Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10. Large numbers use positive powers (75000 = 7.5 × 10⁴) and small numbers use negative powers (0.0003 = 3 × 10⁻⁴). The power of 10 tells which number is larger.
Example 1: Write 0.00045 in standard form.
4.5 × 10⁻⁴.
Example 2: Write 6,000,000 in standard form.
6 × 10⁶.
Quick recap
- Standard form: a × 10ⁿ with 1 ≤ a < 10.
- Large numbers → positive power; small numbers → negative power.
✓ Quick check
Write 75000 in standard form.
75000 = 7.5 × 10⁴.
What is 3 × 10⁻² as an ordinary number?
3 × 0.01 = 0.03.
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