Question

How do you simplify `(-2mn^2)^-3/(4m^-6 n^4)`?

Answer 1

`= color(blue)(-m^3.n^-10)/32`

Explanation:

`(-2mn^2)^-3/(4m^-6 n^4)`

Simplifying the numerator:
`(-2^1m^1n^2)^color(red)(-3`
`=-2^(1 . color(red)(-3)) . m^(1. color(red)(-3)) . n^(2 . color(red)(-3)^`
`= -2^-3m^-3n^-6`
`= color(blue)(-1/2^3m^-3n^-6`
`= color(blue)(-m^-3n^-6)/8`

The expression can now be written as :
`= color(blue)(-m^-3n^-6)/(8.4m^-6 n^4)`

`= color(blue)(-m^-3n^-6)/(32m^-6 n^4)`

Note : `color(blue)(a^m /a^n = a^(m-n)`
Applying the above to the exponents of `m` and `n`

`= (-m^(-3+6).n^(-6-4))/32`
`= color(blue)(-m^3.n^-10)/32`