Question

How do you simplify `\frac { ( - 2x y ^ { - 3} ) ^ { - 5} } { ( x y ^ { - 3} ) ^ { - 3} }`?

Answer 1

`=-y^6/(32x^2)`

Explanation:

There are several laws of indices which can be applied.
One law is not better or stronger than another.

Use the ones you find easiest to recognise and apply.

I like to get get rid of any negative indices first, then simplify.

Recall: `x^color(red)(-m) = 1/x^color(red)(m) " and " 1/y^color(blue)(-n) = y^color(blue)(n)`

#((-2xy^-3)^color(red)(-5))/((xy^-3)^color(blue)(-3)) = ((xy^-3)^color(blue)(3))/((-2xy^-3)^color(red)(5))#

Recall: `(x^m)^n = x^(mxxn)" "` (removes brackets)

`((xy^-3)^color(blue)(3))/((-2xy^-3)^color(red)(5)) =(x^3y^-9)/((-2)^5x^5y^-15)`

`= (x^3y^15)/(-32x^5y^9)" "larr` positive indices

`=-y^6/(32x^2)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
You might like to get rid of the brackets first.

`((-2xy^-3)^color(red)(-5))/((xy^-3)^color(blue)(-3)) = ((-2)^color(red)(-5)x^color(red)(-5)y^color(red)(15))/(x^color(blue)(-3)y^color(blue)(9))`

Then subtract the indices of like bases:

`=(-32^-1)x^(-5+3))y^6`

and deal with the negative indices last:

`= y^6/(-32x^2)`