In many applications of mathematics, we can express numbers as a product of certain number of factors all of which are the same. For example,

4 × 4 = 16; 7 × 7 × 7 = 343; and 7 × 7 × 7 × 7 = 2,401.

Mathematics use the idea of indices to represent a product of a certain number involving the same factor. Indices provide a compact algebraic notation for repeated multiplication. For example,

3 × 3 × 3 × 3 × 3 × 3 = 3⁶.

Indices are used to show numbers that have been multiplied by themselves. The index of a number says how many times to use the number in a multiplication. The plural of index is indices. For example, in 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5. The number 5 is written 8 times and can be written as 5⁸. Here, 5 is called the “base”, 8 is called the index or exponent and 5⁸ is called the power.

Example

a² = a × a
a³ = a × a × a
In general, for any real number a and n is a positive integer,
aⁿ = a × a × a × a × ...n times .

Definition

If a is any real number and n is any natural number, then the n^th power of a, denoted by aⁿ, read as “ a to the power of n” is defined as:

In the expression aⁿ, a is called the base, n is called the index or exponent and aⁿ is the power.

Note that, in the expression (−4)⁶ the base is −4 but in the expression −4⁶ the base is only 4.