4p5 is a short way of writing 4 × p × p × p × p × p.
Indices (plural of index) obey certain rules when they are being multiplied and divided.
Index of one
A number or variable to an index of 1, is equal to itself.
When multiplying numbers or variables with indices, add the indices together.
The bases must be the same number or variable.
When dividing numbers or variables with indices, subtract the indices. The bases must be the same number or variable.
When raising a number or variable with an index to another index, multiply the two indices together.
e.g. (a3)2 = a3x2 = 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 = a3 − 3 = a0 and a3⁄a3 = 1
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 indces are used to show roots or surds.
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