Index (Power) — Definition, Formula & Examples

The index, also called the power or exponent, is the small raised number that tells you how many times to multiply the base by itself. For example, in

5^3

, the index is 3, meaning you multiply 5 three times:

5 \times 5 \times 5

Given a base $a$ and a positive integer $n$ , the expression $a^n$ denotes $a$ multiplied by itself $n$ times. The value $n$ is called the index (or power/exponent) of the expression.

Key Formula

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

Where:

$a$ = The base — the number being multiplied
$n$ = The index (power) — how many times the base is used as a factor

Worked Example

Problem: Evaluate

2^5

Identify the parts: The base is 2 and the index is 5, so you multiply 2 by itself 5 times.

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

Multiply: Work left to right:

2 \times 2 = 4

4 \times 2 = 8

8 \times 2 = 16

16 \times 2 = 32

2^5 = 32

Answer:

2^5 = 32

Why It Matters

The index connects directly to roots: in

\sqrt[n]{a}

, that small

n

is also called the index of the radical, and it reverses the effect of raising to the

n

th power. Understanding indices is essential for simplifying radical expressions and solving equations in algebra.

Common Mistakes

Mistake: Confusing the index of a power with the index of a radical. In

a^3

the index 3 means "multiply three times," but in

\sqrt[3]{a}

the index 3 means "find the cube root."

Correction: Remember that these are inverse operations. Raising to a power and taking a root undo each other, but the index plays a different role in each notation.