BestMaths Online

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

Summary of Rules of Indices

Rules of Indices

Expressions involving indices

 

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

Summary

 

Rules of indices	Description
ax × ay = ax + y	Multiplying (add the indices)
ax⁄ay = ax−y	Dividing (subtract the indices)
(ax ) y = axy	Powers (multiply the indices)
a1 = a	Power of one (stays same)
a0 = 1 , a ≠ 0	Power of zero (always equals one)
a-x = 1⁄ax, 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.

Multiplication

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

Division

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.

because (a³ )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 ^a3⁄_a³ = a^{3 − 3} = a⁰ and ^a3⁄_a³ = 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. 9^x can be written as (3²)^x = 3^2x

example 1	example 2	example 3	example 4