Calculating with Indices

## Calculating with Indices

There are three rules to remember - known as the Laws of Indices.

They only apply when working with the same base numbers.

The laws deal with raising base numbers to 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))

There is a similar law for 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)).

## Example 1

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

93 x 34

9 can be changed to 32, so that the base numbers are the same

= (32)3 x 34

= 3(3x2 + 4)

= 310

## Example 2

What is 45 ÷ 45?

4^5 ÷ 4^5

= frac(4xx4xx4xx4xx4)(4xx4xx4xx4xx4)

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