All About Numbers
Laws of Exponents – The Rules That Make Powers Work
Once numbers start being multiplied and divided as powers, a handful of reliable shortcuts – the laws of exponents – let you combine and simplify them without ever expanding out the repeated multiplication by hand. These five rules, covered one at a time over the next five pages of this section, are used constantly throughout algebra, from simplifying expressions to solving equations with unknowns in the exponent.
The Five Core Laws
| Law | Rule | Example |
|---|---|---|
| Product Law | xa × xb = xa+b | x³ × x² = x⁵ |
| Quotient Law | xa ÷ xb = xa−b | x⁵ ÷ x² = x³ |
| Power of a Power Law | (xa)b = xab | (x²)³ = x⁶ |
| Zero Exponent Law | x⁰ = 1 | 7⁰ = 1 |
| Negative Exponent Law | x−a = 1/xa | 2⁻³ = 1/8 |
Identifying a Law in Action
Which law does x⁵ ÷ x² = x³ demonstrate?
Dividing powers with the same base subtracts the exponents: this is the Quotient Law.
Applying the Laws Together
Simplify 3⁴ × 3² ÷ 3³ and evaluate.
Combine exponents: 4 + 2 − 3 = 3, so the expression equals 3³ = 27.
Real-Life Application
- Algebra: simplifying expressions with variables raised to powers.
- Computer science: analysing how algorithm running time scales.
- Science: combining measurements expressed in scientific notation.
Key Takeaways
- The five laws of exponents let you combine powers without expanding them fully.
- Multiplying powers adds exponents; dividing powers subtracts them.
- Each law is explored in full detail, with worked examples, over the next several pages.
Practice: Laws of Exponents
Identify the Law