Multiplication
a
m
a
n
=a
m+n
Add the indices when the base is the same.
🎯 Learning Objectives
- multiply and divide powers with the same base;
- raise a power to another power;
- apply powers to products and quotients.
Introduction
This lesson develops Laws of Indices from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
The Three Main Laws
Division
a
m
÷a
n
=a
m−n
Subtract the indices when the base is the same.
Power of a power
(a
m
)
n
=a
mn
Multiply the indices.
Exam tip:
The bases must be the same before you add or subtract indices.
✏️ Why does the multiplication law work?
a 3 ×a 2 =(a×a×a)(a×a)=a 5
There are five factors of a , so 3+2=5 .
🧩 Law Builder
Common mistake:
a
3
+a
3
=2a
3
, not
a
6
. The multiplication law does not apply to addition.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
- Simplify x 7 ×x 4 .
- Simplify p 9 ÷p 3 .
- Simplify (m 4 ) 3 .
- Simplify (2ab) 3 .
Show answers
- x 11
- p 6
- m 12
- 8a 3 b 3
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.