Reading an index
In 5⁴, 5 is the base and 4 is the index (or exponent). It means four factors of 5:
5⁴ = 5 × 5 × 5 × 5 = 625
It does not mean 5 × 4.
Index Notation and Powers of 10
Think first: How can one small superscript replace a multiplication containing twenty identical factors?
Open practice worksheet
Know
- base, index, exponent and power
- expanded form and index form
- powers of 10
Understand
- the index counts factors, not multiplication by the index
- why powers of 10 control place value
Can do
- move between expanded and index notation
- evaluate positive-integer powers
- use index notation in prime products and calculations
Powers of 10
For positive whole-number indices, 10ⁿ is 1 followed by n zeros: 10¹ = 10, 10² = 100, 10³ = 1,000. This connects index notation directly to place value.
Indices inside calculations
Apply indices before multiplication, division, addition and subtraction unless brackets change the order.
3 + 2⁴ × 5 = 3 + 16 × 5 = 83
Worked examples
Worked example 1: Expand and evaluate 3⁵
3⁵ = 3 × 3 × 3 × 3 × 3 = 243.
Worked example 2: Write 2 × 2 × 3 × 3 × 3 × 7 in index form
Group equal prime factors: 2² × 3³ × 7. A single factor does not need an index of 1.
Worked example 3: Evaluate 4 × 10³ + 7 × 10
4 × 1,000 + 7 × 10 = 4,000 + 70 = 4,070.
Check your understanding
Reason and apply
1. Write 6⁴ in expanded form and evaluate it.
Model response: 6 × 6 × 6 × 6 = 1,296.
2. Write 2 × 2 × 2 × 5 × 5 in index form.
Model response: 2³ × 5².
3. Evaluate 5 + 3² × 4.
Model response: Indices first: 3² = 9. Then 9 × 4 = 36 and 5 + 36 = 41.
4. Explain the difference between 4³ and 4 × 3.
Model response: 4³ means 4 × 4 × 4 = 64, while 4 × 3 = 12. An index counts repeated factors.
5. Describe 5³ using the terms base, index, exponent and power.
Model response: The base is 5. The index, also called the exponent, is 3. The power 5³ means three factors of 5 and has value 125.