Base
In 3 4 , the base is 3.
🎯 Learning Objectives
- identify the base and index;
- write repeated multiplication in index form;
- evaluate simple powers, including index zero.
Introduction
This lesson develops Meaning 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.
What does an index mean?
An index tells us how many times the base is used as a factor.
a
n
= a × a × ··· × a (n factors)
Index
The index is 4, so 3 is multiplied by itself four times.
Value
3 4 =3×3×3×3=81 .
✏️ Worked Example
Write 5×5×5×5×5 in index form.
The repeated factor is 5, so the base is 5.
There are five factors, so the index is 5.
Answer:
5
5
.
🎚️ Exponent Builder
Choose a base and index.
Common mistake:
2
4
means
2×2×2×2
, not
2×4
.
🧠 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
- Write 7×7×7 in index form.
- Expand 4 3 .
- Evaluate 10 3 .
Show answers
- 7 3
- 4×4×4
- 1000
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.