Laws of Indices

What You'll Learn

Master the complete set of index laws (also called laws of exponents or power rules) essential for GCSE Mathematics. These laws allow you to simplify and manipulate expressions involving powers efficiently.

Why This Matters for GCSE

Exam Coverage: Index laws appear in both Foundation and Higher tier papers, particularly in non-calculator Paper 1. They're essential for:

Simplifying algebraic expressions (Section 4)
Solving exponential equations (Section 5)
Working with standard form (Chapters 1-7 and 1-8)
Surds and advanced number work (Higher tier)
Graph transformations and functions (Section 7)

GCSE Command Words: "Simplify", "Write as a single power", "Evaluate", "Show that"

The 4-Step Method

1

Identify the Law

Determine which index law(s) apply: multiplication, division, power of power, or special cases.

2

Apply the Rule

Use the correct law: add powers (×), subtract powers (÷), or multiply powers (power of power).

3

Simplify

Combine the powers and simplify. Check for further simplification opportunities.

4

Check Special Cases

Apply zero power, negative power, or fractional power rules if needed.

The Complete Index Laws

Law 1: Multiplication Rule

a^m × aⁿ = a^m+n

When multiplying powers with the same base, add the indices

Law 2: Division Rule

a^m ÷ aⁿ = a^m-n

When dividing powers with the same base, subtract the indices

Law 3: Power of a Power

(a^m)ⁿ = a^mn

When raising a power to another power, multiply the indices

Law 4: Zero Power

a⁰ = 1

Any number (except 0) raised to the power 0 equals 1

Law 5: Negative Power

a^-n = 1/aⁿ

A negative power means "one over" that positive power

Law 6: Fractional Power

a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m

A fractional power represents roots (denominator) and powers (numerator)

Worked Examples

Example 1: Multiplication Rule (Foundation)

Simplify: 3⁴ × 3²

Same base (3), so add the powers: 4 + 2 = 6

3⁴ × 3² = 3⁶

3⁶ = 729

Answer: 3⁶ = 729

Mark Scheme: 1 mark for a^m × aⁿ = a^m+n, 1 mark for correct answer

Example 2: Division Rule (Foundation)

Simplify: 5⁷ ÷ 5³

Same base (5), so subtract the powers: 7 - 3 = 4

5⁷ ÷ 5³ = 5⁴

5⁴ = 625

Answer: 5⁴ = 625

Mark Scheme: 1 mark for a^m ÷ aⁿ = a^m-n, 1 mark for correct answer

Example 3: Power of a Power (Foundation & Higher)

Simplify: (2³)⁴

Power raised to a power, so multiply: 3 × 4 = 12

(2³)⁴ = 2¹²

2¹² = 4096

Answer: 2¹² = 4096

Mark Scheme: 1 mark for (a^m)ⁿ = a^mn, 1 mark for correct answer

Example 4: Negative Powers (Higher)

Evaluate: 2^-3

Negative power means "one over" the positive power

2^-3 = 1/2³

= 1/8

Answer: 1/8 or 0.125

Mark Scheme: 1 mark for a^-n = 1/aⁿ, 1 mark for correct answer

Example 5: Combined Laws (Higher)

Simplify: (3⁵ × 3²) ÷ 3⁴

First, multiply: 3⁵ × 3² = 3⁵⁺² = 3⁷

Then divide: 3⁷ ÷ 3⁴ = 3^7-4 = 3³

3³ = 27

Answer: 3³ = 27

Mark Scheme: 1 mark for multiplication, 1 mark for division, 1 mark for final answer

Example 6: Fractional Powers (Higher)

Evaluate: 64^1/3

Fractional power 1/n means the nth root

64^1/3 = ³√64

What number cubed gives 64? 4 × 4 × 4 = 64

Therefore ³√64 = 4

Answer: 4

Mark Scheme: 1 mark for recognizing cube root, 1 mark for correct answer

Practice Questions

Answer these GCSE-style questions. Type your answer and click "Check Answer".

1 [2 marks]

Simplify: 7³ × 7⁵

2 [2 marks]

Simplify: 10⁹ ÷ 10⁴

3 [2 marks]

Simplify: (4²)³

4 [1 mark]

Evaluate: 9⁰

5 [2 marks]

Write as a single power: 2⁶ × 2^-2

6 [2 marks]

Evaluate: 5^-2

7 [3 marks]

Simplify fully: (3⁴ × 3³) ÷ 3⁵

8 [2 marks]

Evaluate: 27^1/3

9 [2 marks]

Evaluate: 16^1/2

10 [3 marks]

Show that: 8^2/3 = 4

Hint: 8^2/3 = (8^1/3)² = (³√8)² = 2² = 4

Common GCSE Misconceptions

❌ WRONG: 2³ × 2⁴ = 2¹² (multiplying the powers)
✅ CORRECT: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ (adding the powers)

❌ WRONG: 5⁰ = 0
✅ CORRECT: 5⁰ = 1 (any number to power 0 is 1)

❌ WRONG: 3^-2 = -9
✅ CORRECT: 3^-2 = 1/3² = 1/9 (negative power means reciprocal)

❌ WRONG: (2³)² = 2⁵ (adding inside bracket)
✅ CORRECT: (2³)² = 2^3×2 = 2⁶ (multiply the powers)

❌ WRONG: 16^1/2 = 16 ÷ 2 = 8
✅ CORRECT: 16^1/2 = √16 = 4 (square root, not division)

GCSE Exam Tips

Non-calculator paper: You MUST know all index laws by heart - no calculator means no shortcuts!
Show your working: Even if you know 2⁶ = 64, write the intermediate step 2³⁺³ = 2⁶ for method marks.
"Write as a single power": Leave as 3⁷, don't calculate 2187 unless asked to "evaluate".
"Show that" questions: You must show every step clearly to prove the given answer (full marks only with complete working).
Negative powers: Always write as a fraction first: 2^-3 = 1/2³ = 1/8 (don't skip steps).
Fractional powers: Remember: denominator = root, numerator = power. So 64^2/3 = (³√64)² = 4² = 16.
Check your base: 2³ × 3⁴ CANNOT be simplified - different bases!

Skills Checklist

By the end of this chapter, you should be able to:

✓ Apply the multiplication rule (a^m × aⁿ = a^m+n)
✓ Apply the division rule (a^m ÷ aⁿ = a^m-n)
✓ Apply the power of a power rule ((a^m)ⁿ = a^mn)
✓ Understand that a⁰ = 1
✓ Work with negative powers (a^-n = 1/aⁿ) [Higher]
✓ Work with fractional powers (a^1/n = ⁿ√a) [Higher]
✓ Combine multiple index laws in one question [Higher]
✓ Solve GCSE-style "show that" problems [Higher]