Laws of Indices Explained — All the Rules with Worked Examples (JAMB/WAEC Mathematics)
The laws of indices are a set of rules that govern how to simplify and calculate expressions involving powers (exponents), covering operations like multiplication, division, and raising a power to another power.
This topic is one of the most fundamental in JAMB and WAEC Mathematics, since it underpins later topics like exponential functions, logarithms, and standard form. This lesson covers every major law with worked examples.
Quick takeaways
- When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
- When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
- When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
- Any non-zero number raised to the power of zero equals 1: a⁰ = 1.
- A negative exponent means “take the reciprocal”: a⁻ⁿ = 1/aⁿ, and a fractional exponent represents a root: a^(1/n) = ⁿ√a.
Timestamps
- 0:00 Introduction
- 0:54 What is an Index?
- 1:03 Law of Multiplication (aᵐ × aⁿ)
- 2:09 Law of Division (aᵐ ÷ aⁿ)
- 2:45 Law of Power of a Power ((aᵐ)ⁿ)
- 3:45 Laws of indices
- 11:40 A working Example
What are indices, and why do they need specific laws?
Indices (also called exponents or powers) represent repeated multiplication of the same number, written as aⁿ, where “a” is the base and “n” is the index. Without specific rules, simplifying expressions involving multiple powers would require expanding everything out fully, which becomes impractical for larger numbers.
The laws of indices provide shortcuts for combining, simplifying, and calculating expressions involving powers efficiently.
What is the multiplication law of indices?
When multiplying two powers with the same base, you add their exponents:
aᵐ × aⁿ = aᵐ⁺ⁿ
For example, 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128. This law only applies when the bases are the same — you cannot directly combine exponents of expressions with different bases this way.
What is the division law of indices?
When dividing two powers with the same base, you subtract the exponent of the divisor from the exponent of the dividend:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
For example, 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625.
What is the power of a power law?
When raising a power to another power, you multiply the exponents together:
(aᵐ)ⁿ = aᵐⁿ
For example, (3²)⁴ = 3²ˣ⁴ = 3⁸. This is different from the multiplication law, and it’s a common point of confusion when a question combines both operations in one expression.
What happens with a zero exponent?
Any non-zero number raised to the power of zero always equals 1:
a⁰ = 1 (for a ≠ 0)
This can seem counterintuitive at first, but it follows logically from the division law: since aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself equals 1, it must follow that a⁰ = 1.
What does a negative exponent mean?
A negative exponent indicates that you should take the reciprocal of the base raised to the corresponding positive exponent:
a⁻ⁿ = 1/aⁿ
For example, 2⁻³ = 1/2³ = 1/8. A negative exponent does NOT make the result negative — it changes the number into a fraction instead, which is one of the most commonly tested misconceptions in this topic.
What does a fractional exponent mean?
A fractional exponent represents a root of the base:
a^(1/n) = ⁿ√a
For example, 8^(1/3) = ³√8 = 2, since 2³ = 8. When the fraction has a numerator other than 1, such as a^(m/n), it represents the nth root of a raised to the mth power: a^(m/n) = (ⁿ√a)ᵐ.
Common mistakes students make with indices
- Adding exponents when multiplying, but mistakenly also adding them when raising a power to a power (which requires multiplying, not adding)
- Treating a negative exponent as making the answer negative, rather than turning it into a reciprocal fraction
- Forgetting that the multiplication and division laws only apply when the bases are the same
- Misapplying the zero exponent rule to 0⁰, which is undefined, rather than only to non-zero bases
Frequently asked questions
Q: What is the rule for multiplying powers with the same base? When multiplying powers with the same base, you add the exponents together: aᵐ × aⁿ = aᵐ⁺ⁿ.
Q: What does a negative exponent mean? A negative exponent means you take the reciprocal of the base raised to the corresponding positive exponent, so a⁻ⁿ = 1/aⁿ — it does not make the result negative.
Q: What is the value of any non-zero number raised to the power of zero? Any non-zero number raised to the power of zero always equals 1.
Q: How do you interpret a fractional exponent? A fractional exponent represents a root of the base, so a^(1/n) equals the nth root of a, and a^(m/n) equals the nth root of a raised to the power of m.
Ready to apply these rules to exponential functions? Read our Exponential Functions lesson →
This is one topic from our full JAMB & WAEC Maths course — see everything covered →
