Curriculum fit: US Algebra 1. Aligned to the Common Core high school algebra standards for expressions (HSA-SSE) and the properties of integer exponents (8.EE.A.1). Check your own state standards and course.
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What Is It?
An exponent tells you how many times to multiply a base by itself. In 5³, the base is 5 and the exponent is 3, so 5³ = 5 × 5 × 5 = 125.
Exponent rules are shortcuts that save you from writing long products. They work when the bases are the same, or when you are raising a product or a power to a power.
Learn the six rules below and practise spotting which rule a question needs.
Key Formulas and Ideas
- Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ.
- Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
- Power of a power: (aᵐ)ⁿ = aᵐⁿ.
- Power of a product: (ab)ⁿ = aⁿbⁿ.
- Zero exponent: a⁰ = 1 (a ≠ 0).
- Negative exponent: a⁻ⁿ = 1 ÷ aⁿ (a ≠ 0).
Worked Examples
Product and Power Rules
Simplify (2x³)² × x⁻⁴.
- Apply the power of a product: (2x³)² = 2² × (x³)² = 4x⁶.
- Multiply by x⁻⁴: 4x⁶ × x⁻⁴ = 4x⁶⁺⁽⁻⁴⁾.
- Add the exponents: 6 − 4 = 2.
✓ Answer 4x²
Quotient Rule With Coefficients
Simplify 6a⁵b² ÷ (3a²b) and write the answer with positive exponents.
- Divide the coefficients: 6 ÷ 3 = 2.
- Subtract the exponents of a: 5 − 2 = 3.
- Subtract the exponents of b: 2 − 1 = 1.
✓ Answer 2a³b
Zero and Negative Exponents
Evaluate 5³ × 5⁻¹ ÷ 5².
- Product: 5³ × 5⁻¹ = 5².
- Quotient: 5² ÷ 5² = 5⁰.
- Any non-zero number to the power 0 is 1.
✓ Answer 1
Common Mistakes to Avoid
- Multiplying the exponents in the product rule. x³ × x⁴ = x⁷, not x¹².
- Forgetting to multiply the coefficients: 2x³ × 3x² = 6x⁵, not 5x⁵.
- Thinking (x + y)² = x² + y². It is not; you must multiply it out.
- Treating a⁰ as 0. Any non-zero number to the power 0 is 1.
- Reading −3² as 9. The exponent applies only to the 3, so −3² = −9, while (−3)² = 9.
Practice Questions
Frequently Asked Questions
Why is any number to the power 0 equal to 1?
Use the quotient rule: x³ ÷ x³ = x⁰, but anything divided by itself is 1. So x⁰ = 1, as long as x is not 0.
What does a negative exponent mean?
A negative exponent means take the reciprocal. So 2⁻³ = 1 ÷ 2³ = 1/8. It does not make the number negative.
Can I use the product rule with different bases?
No. The rule x³ × x⁴ = x⁷ needs the same base. For different bases, such as 2³ × 3², work out each power first.
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Reviewed by StudyMorrow editorial review on 6 Nov 2026.