What is the difference between the index and the base?

The base is the full-size number being multiplied, and the index is the small raised number that tells you how many times to use the base as a factor. In $6^4$, the base is 6 and the index is 4, so you calculate 6 × 6 × 6 × 6 = 1296.

Index Notation (Powers) — Definition, Formula & Examples

Index notation is a shorthand way of writing repeated multiplication of the same number, using a small raised number (the index or exponent) to show how many times the base is multiplied by itself. For example, instead of writing 3 × 3 × 3 × 3, you write 3⁴.

In index notation, an expression $a^n$ represents the product of $n$ identical factors of $a$ , where $a$ is the base and $n$ is the index (also called the exponent or power). For any positive integer $n$ , this means $a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ factors}}$ .

Key Formula

a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ factors}}

Where:

$a$ = The base — the number being multiplied repeatedly
$n$ = The index (exponent) — how many times the base appears as a factor

How It Works

To use index notation, identify the number being repeatedly multiplied (the base) and count how many times it appears as a factor (the index). Write the base in normal size and the index as a small superscript to the right. When you read

5^3

, you say "five to the power of three" or "five cubed." Index notation keeps expressions compact — writing

2^{10}

is far easier than writing out ten 2s multiplied together. You can also work backwards: given

7^2

, expand it as

7 \times 7 = 49

Worked Example

Problem: Write 2 × 2 × 2 × 2 × 2 in index notation, then evaluate it.

Identify the base: The number being repeated is 2.

\text{base} = 2

Count the factors: The number 2 appears 5 times in the product.

\text{index} = 5

Write in index notation: Place the index as a superscript on the base.

2^5

Evaluate: Multiply step by step: 2 × 2 = 4, × 2 = 8, × 2 = 16, × 2 = 32.

2^5 = 32

Answer:

2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32

Another Example

Problem: Evaluate

4^3

Expand: The index 3 tells you to write the base 4 as a factor three times.

4^3 = 4 \times 4 \times 4

Multiply: First multiply 4 × 4 = 16, then 16 × 4 = 64.

4 \times 4 \times 4 = 64

Answer:

4^3 = 64

Visualization

Why It Matters

Index notation is a core skill in middle school maths that reappears throughout algebra, science, and computing. In GCSE and high school courses, you need it to simplify expressions, work with the laws of indices, and handle scientific notation for very large or very small numbers. Engineers and programmers rely on powers constantly — for example, computer memory is measured in powers of 2 (like

2^{10} = 1024

bytes in a kilobyte).

Common Mistakes

Mistake: Multiplying the base by the index instead of using repeated multiplication (e.g., writing 3⁴ = 3 × 4 = 12).

Correction: The index tells you how many times to multiply the base by itself. So 3⁴ = 3 × 3 × 3 × 3 = 81, not 3 × 4.

Mistake: Confusing

a^2

with

2a

(e.g., thinking 5² = 10).

Correction:

5^2

means 5 × 5 = 25. The expression

2 \times 5 = 10

is a completely different operation. The raised 2 means "squared," not "times 2."

Related Terms

Exponent — Another name for the index in index notation
Power — The result of raising a base to an index
Exponentiation — The operation that index notation represents
Product — The result of the repeated multiplication
Arithmetic — The broader category of basic number operations
Formula — Index notation appears inside many formulas
Multiplicative Inverse of a Number — Connected through negative index notation