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))`.**

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

9^{3} x 3^{4}

9 can be changed to 3^{2}, so that the base numbers are the same

= (3^{2})^{3} x 3^{4}

= 3^{(3x2 + 4)}

= 3^{10}

Answer: 3^{10}

What is 4^{5} ÷ 4^{5}?

`4^5 ÷ 4^5`

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

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

Answer: 1

See also Terms with Powers

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