Zero index
a
0
=1
for a≠0 .
🎯 Learning Objectives
- use the zero-index law;
- rewrite negative indices as reciprocals;
- interpret fractional indices as roots and powers.
Introduction
This lesson develops Zero, Negative & Fractional 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.
Visual Exploration
Notice→Connect→Explain
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
Zero and Negative Indices
Negative index
a
−n
=
1
a
n
A negative index means reciprocal, not a negative value.
2 3 , 2 2 , 2 1 , 2 0 gives 8, 4, 2, 1. Each step divides by 2.
Fractional Indices
a
1/n
=ⁿ√a and a
m/n
=ⁿ√(a
m
)=(ⁿ√a)
m
| Expression | Meaning | Value |
|---|---|---|
| 25 1/2 | Square root of 25 | 5 |
| 27 1/3 | Cube root of 27 | 3 |
| 16 3/4 | Fourth root, then cube | 8 |
Helpful order:
The denominator gives the root; the numerator gives the power.
✏️ Worked Example
Evaluate 64 2/3 .
The denominator 3 means take the cube root:
∛64=4
.
The numerator 2 means square:
4
2
=16
.
Therefore
64
2/3
=16
.
🧠 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
- Evaluate 9 0 .
- Write x −4 with a positive index.
- Evaluate 81 1/2 .
- Evaluate 32 2/5 .
Show answers
- 1
- 1/x 4
- 9
- 4
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.