The *Laws of Indices* which apply to numbers are also applied to algebra. Consider `y^3 xx y^2`.

The term `y^3` can be expanded to `y xx y xx y`. The other term, `y^2`, can be expanded to `y xx y`. Therefore `y^3 xx y^2 = y xx y xx y xx y xx y`, which is equal to `y^5`.

This can be written as `y^a xx y^b = y^(a + b)`. When multiplying the same variable, add the powers. Note that the variable letter must be the same for both the terms being multiplied.

Division works in a similar way. The rule is written as `y^a ÷ y^b = y^(a - b)`. When dividing the same variable, subtract the powers. The example `e^6 ÷ e^3` can be shown as:

`frac(e^6)(e^2)`

`= frac(e xx e xx e xx e xx e xx e)(e xx e)`

`= frac(e xx e xx e xx e)(1)`

`= e^4`

The division rule leads to the fact that `e^0 = 1`. This can be shown by `e^2 ÷ e^2`

`= frac(e xx e)(e xx e)`

`= frac(e)(e)` (divided top and bottom by `e`)

`= 1` (divided top and bottom by `e` again)

which leads to the fact that any value to the power of zero equals 1.

Simplify `x^3 xx 4x^3`

`x^3 xx 4x^3`

`= 4 xx x^(3 + 3)`

`= 4x^6`

Answer: `4x^6`

Simplify `4a^5b ÷ 2a^3`

Calculate the numbers: `4 ÷ 2 = 2`

Calculate the variable `a: a^5 ÷ a^3 = a^(5-3) = a^2`

The variable `b` is not divided

Put it all together: `2a^2b`

Answer: `2a^2b`

See also Calculating with Indices

Check out our iOS app: tons of questions to help you practice for your GCSE maths. Download free on the App Store (in-app purchases required).