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))`.

1. Calculate 9^{3} x 3^{4}. Give your answer as a power of 3.

Answer: 3^{10}

9^{3} x 3^{4}
= (3^{2})^{3} x 3^{4}
= 3^{(3x2 + 4)}
= 3^{10}

2. What is 4^{5} ÷ 4^{5}?

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.

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