Rules of Indices – Complete IGCSE Guide with Examples

Why Indices Matter at IGCSE

Indices (also called powers or exponents) are one of the most fundamental topics in IGCSE Mathematics. They appear in algebra, number work, standard form, growth and decay problems, and equation solving. Both Core and Extended students need a solid grasp of the rules, though Extended candidates face more demanding applications such as solving equations involving fractional and negative indices.

Getting the rules right is essential because index errors cascade through a solution, often costing multiple marks on a single question.

The Basic Rules

Rule 1: Multiplication — Add the Powers

When multiplying terms with the same base, add the indices.

aᵐ × aⁿ = aᵐ⁺ⁿ

Example: x³ × x⁵ = x³⁺⁵ = x⁸

Example: 2⁴ × 2³ = 2⁷ = 128

Important: The bases must be the same. You cannot simplify x³ × y⁵ using this rule.

Rule 2: Division — Subtract the Powers

When dividing terms with the same base, subtract the indices.

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example: x⁷ ÷ x² = x⁷⁻² = x⁵

Example: 5⁶ ÷ 5⁴ = 5² = 25

Rule 3: Power of a Power — Multiply the Powers

When raising a power to another power, multiply the indices.

(aᵐ)ⁿ = aᵐⁿ

Example: (x³)⁴ = x¹²

Example: (2³)² = 2⁶ = 64

Rule 4: Power of Zero

Any non-zero number raised to the power of zero equals 1.

a⁰ = 1 (provided a ≠ 0)

This follows logically from the division rule: a³ ÷ a³ = a⁰ = 1.

Example: 7⁰ = 1

Example: (3x)⁰ = 1

Negative Indices

A negative index means “take the reciprocal.”

a⁻ⁿ = 1/aⁿ

Example: x⁻² = 1/x²

Example: 5⁻¹ = 1/5

Example: (2/3)⁻² = (3/2)² = 9/4

Worked Example 1: Simplify 4⁻² × 2³.

First, write 4 as 2²: (2²)⁻² × 2³ = 2⁻⁴ × 2³ = 2⁻¹ = 1/2.

This technique of converting to a common base is extremely useful and appears regularly in exam questions.

Fractional Indices

Fractional indices combine roots and powers.

a^(1/n) = ⁿ√a (the nth root of a)

a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

Example: 8^(1/3) = ³√8 = 2

Example: 27^(2/3) = (³√27)² = 3² = 9

Example: 16^(3/4) = (⁴√16)³ = 2³ = 8

The Order Matters (Practically)

Mathematically, (ⁿ√a)ᵐ and ⁿ√(aᵐ) give the same answer. However, it is almost always easier to take the root first and then raise to the power. Taking the root first keeps the numbers small and manageable.

Worked Example 2: Evaluate 32^(−2/5).

Step 1: Deal with the fraction. The denominator 5 means fifth root. ⁵√32 = 2.

Step 2: Deal with the numerator. Raise to the power 2. 2² = 4.

Step 3: Deal with the negative sign. Take the reciprocal. 1/4.

Answer: 32^(−2/5) = 1/4

Applying the Rules to Algebraic Expressions

Worked Example 3: Simplify (3x²y³)² × (2xy⁴).

Step 1: Apply the power to the bracket. (3x²y³)² = 9x⁴y⁶.

Step 2: Multiply. 9x⁴y⁶ × 2xy⁴ = 18x⁵y¹⁰.

Worked Example 4: Simplify (8x⁶)^(2/3).

Step 1: Apply the fractional index to each factor. 8^(2/3) × (x⁶)^(2/3).

Step 2: 8^(2/3) = (³√8)² = 2² = 4.

Step 3: (x⁶)^(2/3) = x⁴.

Answer: 4x⁴

Solving Equations with Indices

Worked Example 5: Solve 2ˣ = 32.

Write 32 as a power of 2: 32 = 2⁵.

So 2ˣ = 2⁵, therefore x = 5.

Worked Example 6: Solve 9ˣ = 27.

Write both sides with base 3: (3²)ˣ = 3³, so 3²ˣ = 3³.

Equate the powers: 2x = 3, so x = 3/2.

Worked Example 7: Solve x^(−1/2) = 4.

Raise both sides to the power −2: [x^(−1/2)]^(−2) = 4^(−2).

x¹ = 1/16.

x = 1/16

Common Mistakes to Avoid

1. Adding indices when you should multiply. Remember: multiplication of terms uses addition of indices, but power of a power uses multiplication.
2. Applying index rules to different bases. 2³ × 3² cannot be simplified using index laws.
3. Forgetting that a negative index means reciprocal, not a negative number. 2⁻³ = 1/8, not −8.
4. Mishandling fractional indices. Always root first, then power — it keeps arithmetic simpler and reduces errors.
5. Ignoring coefficients. In (3x²)³, the 3 is also cubed, giving 27x⁶, not 3x⁶.

Quick Reference Summary

Rule	Formula
Multiply	aᵐ × aⁿ = aᵐ⁺ⁿ
Divide	aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of power	(aᵐ)ⁿ = aᵐⁿ
Zero power	a⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ
Fractional index	a^(m/n) = (ⁿ√a)ᵐ

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