Index Notation
Understand powers, squared and cubed numbers, evaluate expressions in index form, and discover the index laws.
What is Index Notation?
Index notation is a shorthand way of writing repeated multiplication. Instead of writing 2 × 2 × 2 × 2, we write 24. The base is the number being multiplied, and the index (or exponent) tells us how many times to multiply it by itself.
Parts of a Power
34 = 3 × 3 × 3 × 3 = 81
52
"5 squared"
5 × 5 = 25
Area of a 5×5 square
43
"4 cubed"
4 × 4 × 4 = 64
Volume of a 4×4×4 cube
25
"2 to the power 5"
2×2×2×2×2 = 32
Any index is possible!
Square & Cube Numbers
Square numbers are made by multiplying a number by itself (n2). Cube numbers are made by multiplying a number by itself twice (n3). These appear regularly in geometry and measurement.
Perfect Squares
1
1²
4
2²
9
3²
16
4²
25
5²
36
6²
49
7²
64
8²
81
9²
100
10²
Perfect Cubes
1
1³
8
2³
27
3³
64
4³
125
5³
216
6³
343
7³
1000
10³
Introduction to Index Laws
Index laws are rules that help us simplify expressions involving powers with the same base. At Year 7, we introduce two key laws:
Multiplication Law
am × an = am+n
When multiplying powers with the same base, add the indices.
32 × 34 = 32+4 = 36 = 729
Check: 9 × 81 = 729 ✓
Division Law
am ÷ an = am−n
When dividing powers with the same base, subtract the indices.
25 ÷ 22 = 25−2 = 23 = 8
Check: 32 ÷ 4 = 8 ✓
Important Reminder:
Index laws only apply when the bases are the same. You cannot simplify 23 × 32 using index laws because the bases are different.
Key Vocabulary
Index (Exponent)
The small raised number that tells you how many times to multiply the base by itself. In 53, the index is 3.
Base
The number being multiplied repeatedly. In 53, the base is 5.
Square Number
The result of multiplying a number by itself (n2). Called "square" because it gives the area of a square with side n.
Cube Number
The result of multiplying a number by itself twice (n3). Called "cube" because it gives the volume of a cube with edge n.
Worked Examples
Evaluate 26.
26 = 2 × 2 × 2 × 2 × 2 × 2
= 4 × 4 × 4 = 16 × 4 = 64
Simplify: 53 × 52
Same base (5), so add the indices: 3 + 2 = 5
53 × 52 = 55 = 3125
Write 32 in index form using base 2.
32 = 2 × 16 = 2 × 2 × 8 = 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2
That is 5 twos multiplied together.
32 = 25
Knowledge Check
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Key Concepts Summary
- ●Index notation an means multiplying the base a by itself n times.
- ●Square numbers end in 1, 4, 5, 6, 9 or 0. Memorise squares up to 122 = 144.
- ●Cube numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- ●Multiplication law: am × an = am+n (add indices when multiplying same base).
- ●Division law: am ÷ an = am−n (subtract indices when dividing same base).