Calculating with Indices

Calculating with Indices

GCSE(F), GCSE(H),

There are three rules to remember - known as the Laws of Indices - when working with two base numbers that have been raised to different powers.

For example:

`2^3 xx 2^4`

`= (2xx2xx2) xx (2xx2xx2xx2) = 2^7`

which is the same as adding the indices.

The law for multiplying is `2^m xx 2^n = 2^((m+n))`

When dividing:

`4^5 ÷ 4^3`

`= frac(4xx4xx4xx4xx4)(4xx4xx4)`

`= 4 xx4 = 4^2`

and in this case the same answer is obtained by subtracting the indices.

The law for dividing: `4^m÷4^n=4^((m-n))`

When raising a number to a further power, the indices are multiplied:

`(5^3)^2` (5 raised to the power of three, then raised to the power of 2)

`(5 xx5 xx 5) xx (5 xx 5 xx 5) = 5^6`

and can be re-written by multiplying the indices.

The law for raised to a power: `(5^m)^n = 5^((mxxn))`.

Examples

1. Calculate 93 x 34. Give your answer as a power of 3.

Answer: 310

93 x 34 = (32)3 x 34 = 3(3x2 + 4) = 310

2. What is 45 ÷ 45?

Answer: 1

`4^5 ÷ 4^5` `= frac(4xx4xx4xx4xx4)(4xx4xx4xx4xx4)` `= frac(1)(1) = 1 = 4^0`. Anything raised to the power of zero is always 1.