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 xx e xx e xx e xx e xx e)(e xx e)`
`= frac(e xx e xx e xx e)(1)`
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)`
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`
See also Calculating with Indices