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📘 GCSE Foundation Maths Rescue System

Unit 3: Indices & Powers

Grade 4/5 Focus · Evaluate, multiply, divide powers and understand index laws

Repeated Multiplication Multiplying Powers Dividing Powers Negative Bases Index Laws
AQA 8300 Edexcel 1MA1 OCR J560
🎯 Unit Progress
0 / 5 skills
How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
GCSE Foundation Maths Revision Pathway

A structured 10-unit revision system to help Foundation students build from Grade 3 towards Grade 4/5.

1 Algebra & Sequences
2 Expanding & Factorising
📖 Unit 3 — Indices & Powers (current)
4 Linear Equations
5 Inequalities
6 Straight-Line Graphs
7 Non-Linear Graphs
8 Real-Life Graphs
9 Graph Applications
10 Coordinate Geometry
🔥
Warm-Up: Retrieval Practice
⏱ Before You Start
Answer from memory — no notes!
Q1  Write 2³ as repeated multiplication
Q2  Simplify:   3² × 3³
Q3  Simplify:   5⁴ ÷ 5²
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: Repeated Multiplication

An index (power) is a shortcut for repeated multiplication.

2 × 2 × 2 × 2
=
2⁴ = 16

The small number (index) tells you how many times to multiply the base by itself. It’s NOT multiplying by the power!

2 4 = 2 × 2 × 2 × 2 = 16 ↑ base ↑ index multiply 2 by itself 4 times
📖
Core Definitions
Base
The number being multiplied repeatedly
Index (Power)
How many times the base is multiplied by itself
Square
A number multiplied by itself (power of 2)
Cube
A number multiplied by itself three times (power of 3)
💡 Why Index Laws Work
When multiplying powers with the same base:
2³ × 2² = (2×2×2) × (2×2) = 2⁵
We are counting total identical factors.
✏️
Worked Examples
Worked Example 01Repeated Multiplication
Write 4³ as repeated multiplication
1
The power (3) tells us how many times to multiply 4 by itself.
✓ Answer
4³ = 4 × 4 × 4
Worked Example 02Evaluating a Power
Evaluate 3²
1
Write as repeated multiplication: 3² = 3 × 3
2
Calculate: 3 × 3 = 9
✓ Answer
9
Worked Example 03Negative Base PowersKEY SKILL
Evaluate: (a) (−2)²    (b) (−2)³
1
(a) (−2)² = (−2) × (−2) = 4  (even power → positive)
2
(b) (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8  (odd power → negative)
✓ Answer
(a) 4    (b) −8
⚠️ Even powers → positive. Odd powers → keep the sign.
Worked Example 04Multiplying Powers (Same Base)
Simplify: 5² × 5³
1
Same base → add the indices: 2 + 3 = 5
✓ Answer
5⁵
💡 The base (5) stays the same; only the indices are added.
Worked Example 05Dividing Powers (Same Base)
Simplify: 7⁵ ÷ 7²
1
Same base → subtract the indices: 5 − 2 = 3
✓ Answer
7³
Worked Example 06Bases Must Match!IMPORTANT
Simplify: 2³ × 3²
1
Bases are different (2 and 3) — cannot use index laws!
2
Evaluate separately: 2³ = 8, 3² = 9
3
Multiply: 8 × 9 = 72
✓ Answer
72
⚠️ Index laws ONLY work when the bases match!
⭐
Special Index Rules
Worked Example 07The Zero Power Rule (a⁰ = 1)EXAM TIP
Evaluate:   7⁰
1
Rule: Any non-zero number to the power of 0 equals 1.
2
Why? Because 7³ ÷ 7³ = 7⁰ (subtract indices), but 7³ ÷ 7³ also = 1 (anything ÷ itself = 1).
✓ Answer
7⁰ = 1
⚠️ This catches many students: 7⁰ = 1, NOT 0 and NOT 7. The base doesn't matter — 100⁰ = 1, (−3)⁰ = 1, x⁰ = 1.
🔢 Power of 1
Any number to the power of 1 stays the same: 6¹ = 6
The Two Index Laws MULTIPLY → ADD indices a³ × a² = a⁵ 3 + 2 = 5 DIVIDE → SUBTRACT indices a⁵ ÷ a² = a³ 5 − 2 = 3
Same base? Multiply → add. Divide → subtract. Base never changes.
✅
Process Checklists
📋 Multiplying Powers
📋 Dividing Powers
🧮
Arithmetic Support
(−2)² = 4
(−2)³ = −8
Add indices when multiplying
Subtract indices when dividing
Rules only work when bases match
💪
Tiered Practice
🔓 How It Works
Tap any card to reveal the full working and answer.
0 / 12
Problems revealed
Tap problems to reveal working and answers
🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Evaluate: 5²
>
🔍
Mistake Detective
⚠️ These mistakes cost marks
Study each one carefully.
❌ Wrong
2³ × 3² = 6⁵
Bases are different — cannot combine
✅ Correct
2³ = 8, 3² = 9 8 × 9 = 72
❌ Wrong
(−3)² = −9
Even power of a negative is positive
✅ Correct
(−3)² = (−3) × (−3) = 9
❌ Wrong
5² × 5³ = 5⁶
Multiplied indices instead of adding
✅ Correct
5² × 5³ = 5⁵ (Add: 2 + 3 = 5)
🎓
Examiner Lens
🎓 To Gain Full Marks
🏆
Mastery Checklist
🏆 Unit 3 Mastery
Nice work — you’ve completed Unit 3!

7 units remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.

Ready for Unit 4?

If you can do these 3 things confidently, you're ready to move on:

👨‍👩‍👧 For Parents

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✔ 170+ practice questions
✔ Step-by-step explanations
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