The three index laws let you multiply, divide and raise powers in algebra without expanding every bracket. For GCSE maths you need: the multiplication law (add the indices), the division law (subtract the indices), and the power law (multiply the indices). All three apply whenever the bases are the same.
What are the three index laws?
An index (plural: indices) is the small raised number that tells you how many times the base is multiplied by itself. The three laws give you shortcuts when the bases match:
| Law | Rule | Example |
|---|---|---|
| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x⁴ = x⁷ |
| Division | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | y⁸ ÷ y³ = y⁵ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (z²)⁵ = z¹⁰ |
These laws apply to letters, numbers and any combination. They do not apply when the bases are different — you cannot simplify x³ × y⁴ any further, because x and y are different bases.
How do you use the multiplication law?
When you multiply two powers with the same base, add the indices.
Worked example 1: Simplify 3a⁴ × 5a²
- Multiply the coefficients: 3 × 5 = 15.
- Apply the multiplication law to the a terms: a⁴ × a² = a⁴⁺² = a⁶.
- Combine: 15a⁶
Worked example 2: Simplify 2x²y³ × 4x⁵y
- Multiply the coefficients: 2 × 4 = 8.
- Apply the law to x terms: x² × x⁵ = x⁷.
- Apply the law to y terms: y³ × y¹ = y⁴ (remember y = y¹).
- Answer: 8x⁷y⁴
How do you use the division law?
When you divide two powers with the same base, subtract the denominator index from the numerator index.
Worked example 3: Simplify 6m⁷ ÷ 2m³
- Divide the coefficients: 6 ÷ 2 = 3.
- Apply the division law: m⁷ ÷ m³ = m⁷⁻³ = m⁴.
- Answer: 3m⁴
What if the result is a negative index? If the denominator index is larger: p² ÷ p⁵ = p²⁻⁵ = p⁻³. A negative index means a reciprocal: p⁻³ = 1/p³.
How do you use the power of a power law?
When a power is raised to another power, multiply the indices.
Worked example 4: Simplify (n³)⁴
Multiply the indices: 3 × 4 = 12. Answer: n¹²
Worked example 5: Simplify (2p²)³
Raise every factor inside the bracket to the power 3:
- 2³ = 8
- (p²)³ = p⁶
- Answer: 8p⁶
A common error is forgetting to cube the coefficient 2 — make sure the power outside applies to everything inside the bracket.
What do the zero and negative index rules mean?
Two special cases follow directly from the division law:
- Zero index: Any non-zero number raised to the power 0 equals 1. Example: 7⁰ = 1, x⁰ = 1. This comes from aⁿ ÷ aⁿ = a⁰ = 1.
- Negative index: a⁻ⁿ = 1/aⁿ. Example: 5⁻² = 1/5² = 1/25.
These are covered in the companion article on negative and fractional indices, but they emerge naturally from the division law you already know.
How do the index laws appear in GCSE exam questions?
GCSE questions rarely test one law in isolation. Expect combinations:
Worked example 6: Simplify (3x²y)² ÷ (9x³)
- Expand the bracket first using the power law: (3x²y)² = 3² × (x²)² × y² = 9x⁴y².
- Now divide: 9x⁴y² ÷ 9x³.
- Coefficients: 9 ÷ 9 = 1.
- x terms: x⁴ ÷ x³ = x¹ = x.
- y remains: y².
- Answer: xy²
What mistakes should you avoid?
| Mistake | What goes wrong | Correct approach |
|---|---|---|
| Adding indices when multiplying different bases | x² × y³ ≠ xy⁵ | Only add indices with matching bases |
| Multiplying indices instead of adding | x³ × x⁴ ≠ x¹² | Add: x³⁺⁴ = x⁷ |
| Forgetting to apply the outer power to the coefficient | (2x³)⁴ ≠ 2x¹² | (2x³)⁴ = 2⁴x¹² = 16x¹² |
| Confusing a⁰ = 0 | 5⁰ ≠ 0 | 5⁰ = 1 (any non-zero base) |
Frequently asked questions
Do the index laws work with fractions as bases?
Yes. The laws apply to any base — whole numbers, fractions or letters. For example, (½)² × (½)³ = (½)⁵ = 1/32. The base stays the same throughout; only the indices change.
Can I use the multiplication law to combine x³ and y³?
No. The multiplication law requires the bases to be the same. x³ × y³ cannot be simplified using the multiplication law — the result stays as x³y³. However, (xy)³ = x³y³, which is a different relationship.
Why does any number to the power 0 equal 1?
Use the division law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But any number divided by itself also equals 1. So a⁰ = 1 for any non-zero a. The case 0⁰ is undefined and does not appear at GCSE.
How do index laws connect to standard form?
Standard form numbers like 3.2 × 10⁵ use powers of 10. When you multiply two standard form numbers, you add the powers of 10 (multiplication law) and combine the coefficients separately. For example, (3 × 10⁴) × (2 × 10³) = 6 × 10⁷.
For guided GCSE algebra practice with Professor Pi — visit aitutors.me.