4p5 is a short way of writing 4 × p × p × p × p × p. 
The 4 is called the 
The p is called the base or variable.
The 5 is called the 
index, power or exponent.


Indices (plural of index) obey certain rules when they are being multiplied and divided.



Rules of indices


ax × ay = ax + y

Multiplying (add the indices)

axay = ax−y

Dividing (subtract the indices)

(ay = axy

Powers (multiply the indices)

a1 = a

Power of one (stays same)

a0 = 1 , a ≠ 0

Power of zero (always equals one)

a-x = 1ax, a ≠ 0
Negative index (take the reciprocal)

Fractional index (the root of a number)


Rules of Indices

Index of one

A number or variable to an index of 1, is equal to itself.

e.g. 41 = 4 
y1 = y



When multiplying numbers or variables with indices, add the indices together.

The bases must be the same number or variable.

e.g. a3 × a4 = a 3 + 4 = a7

because a3 × a4 = (a × a × a) × (a × a × a × a) = a7


When dividing numbers or variables with indices, subtract the indices. The bases must be the same number or variable.


Two indices

When raising a number or variable with an index to another index, multiply the two indices together.

e.g. (a3)= a3x2 = a6

because (a)2 = (a x a x a) x (a x a × a) = a6

Index of zero

Any number or variable to the index of 0 is equal to 1. Note: 00 is undefined. i.e. it does not exist.

e.g. a0 = 1

because a3a3 = a3 − 3 = a0 and a3a3 = 1

Negative index

A number or variable raised to a negative index is equal to the reciprocal of the number or variable to the same positive index.


Fractional index

Fractional indces are used to show roots or surds.

example 1
example 2



Expressions involving Indices

To simplify algebraic expressions involving indices, use the rules of indices.

Simplify the signs first, numbers next and then the similar variables.

For problems with bases that are numbers, these bases must be the same. e.g. 9x can be written as (32)x = 32x

example 1
example 2
example 3
example 4