Question

How do you get rid of negative exponents in `(\frac { x ^ { 4} y ^ { - 2} } { x ^ { - 3} y ^ { 5} } ) ^ { - 1}`?

Answer 1

`y^7/x^7`

Explanation:

First, the -1 as the exponent of the whole fraction can simplify down to the reciprocal of the fraction itself.
`(x^-3y^5)/(x^4y^-2)`
then you can subtract the exponents in the denominator from the exponents in the numerator. This will give you
`x^-7y^7`
`x^-7=1/x^7` so
`(1/x^7)y^7`
`y^7/x^7`

Answer 2

`y^7/x^7`

Explanation:

Given: `((x^4y^-2)/(x^-3y^5))^-1`

The key to this one is understanding that a negative exponent means division so the result will be a reciprocal. That means to flip the numerator 'top' and denominator 'bottom'.

Since the exponent outside the bracket is `-1` we can clear it right away by flipping the whole equation to end up with a new one with exponent `1` which means multiplied by `1`:

`((x^4y^-2)/(x^-3y^5))^-1 = 1* (x^-3y^5)/(x^4y^-2)`

But `x^-3 = 1/x^3`; and `y^-2 = 1/y^2`; so:

`(x^-3y^5)/(x^4y^-2) = (y^5/x^3)/(x^4/y^2)` which we will need to invert and multiply:

`(x^-3y^5)/(x^4y^-2) = (y^5/x^3)*(y^2/x^4) = y^7/x^7`