Index laws are often memorised as rules about powers. They become much easier to trust when each law is traced back to repeated multiplication.
The key idea is that an index records how many copies of a base are multiplied together. The laws simply keep track of those copies when expressions are multiplied, divided or raised to another power.
Meaning of a power
a⁴ = a × a × a × aThe base is a. The index 4 records four factors of a.
Product law
a³ × a⁴
= (a×a×a)(a×a×a×a)
= a⁷Therefore:
aᵐ × aⁿ = aᵐ⁺ⁿThe exponents add because the number of identical factors adds.
Quotient law
a⁷ ÷ a³ = a⁴Three matching factors cancel, leaving four. Therefore:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a ≠ 0Power of a power
(a³)⁴ = a³ × a³ × a³ × a³ = a¹²Hence:
(aᵐ)ⁿ = aᵐⁿThe indices multiply because the group of m factors is repeated n times.
Zero index
Using the quotient law:
a³ ÷ a³ = a³⁻³ = a⁰But any non-zero number divided by itself is 1. Therefore a⁰ = 1 for a ≠ 0.
Negative indices
a² ÷ a⁵ = a⁻³Cancel two factors from numerator and denominator and three a factors remain below:
a⁻³ = 1/a³A negative index does not make the value negative. It indicates a reciprocal power.
Fractional indices
To preserve the power-of-a-power law, a^(1/2) must be a number whose square is a:
(a^(1/2))² = a¹ = aSo, for appropriate real a, a^(1/2) = √a. More generally:
a^(1/n) = ⁿ√a
a^(m/n) = ⁿ√(aᵐ)Worked example 1
x⁵ × x³ ÷ x² = x^(5+3−2) = x⁶Worked example 2
(2x³)² = 2² × x⁶ = 4x⁶The outer power applies to the coefficient and to the variable factor.
Worked example 3
27^(2/3)
= (³√27)²
= 3²
= 9Different bases require caution
The product law aᵐ × aⁿ = aᵐ⁺ⁿ requires the same base. For example, 2³ × 3² cannot become 6⁵.
Instead calculate each factor or rewrite to a common base only when mathematically valid.
Addition is different
x² + x³ cannot be simplified to x⁵. Index laws describe multiplication and division of powers, not addition of unlike terms.
Indices and prime factorisation
Prime factorisation is index notation applied to numbers. For example:
360 = 2³ × 3² × 5This connection is why index laws, HCF, LCM and standard form belong to the same structural family.
Common errors
Adding indices across addition. x² + x³ stays as written unless another algebraic factorisation is possible.
Multiplying indices when multiplying powers. x² × x³ = x⁵, not x⁶.
Forgetting coefficient powers. (3x²)² = 9x⁴.
Treating a negative index as a negative number. x⁻² = 1/x².
Applying same-base laws to different bases. Check the base before choosing a law.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| x²×x³ = x⁶ | Product law meaning | Expand repeated factors once |
| x⁻³ = −x³ | Negative index misconception | Derive through quotient cancellation |
| (2x)³ = 2x³ | Outer power distribution | Apply power to every factor |
| x²+x³ = x⁵ | Operation confusion | Separate addition from multiplication |
Practice
- Simplify a⁷ × a⁴.
- Simplify b⁹ ÷ b³.
- Simplify (x⁴)³.
- Write y⁻⁵ with a positive index.
- Evaluate 16^(3/4).
Answers
1. a¹¹. 2. b⁶. 3. x¹². 4. 1/y⁵. 5. 8.
Connected routes
Use Prime Factors as the Hidden Structure of Whole Numbers for numerical decomposition and Standard Form for powers of ten in scale and scientific notation. Return to the Mathematics Learning Hub.