Question

How do you simplify `((3^-2 x^5 y^-5)^10 (3^3 x^-5 y^2)^3)/((3^5 x^-3 y^4)^2)`?

Answer 1

`( x^41 )/(3^21 y^52`

Explanation:

Simplify

`((3^-2 x^5 y^-5)^10  (3^3 x^-5 y^2)^3)/((3^5 x^-3  y^4)^2)`

This problem isn't hard conceptually; it's just painstaking.

1) Clear the parentheses by raising all the powers inside the parentheses by the powers outside.
To raise a power to a power, you multiply.

`(3^-20  x^50  y^-50  3^9  x^-15  y^6)/(3^10  x^-6  y^8`

2) Clear the negative exponents by flipping them to the opposite side of the fraction bar and reversing the negative signs to positive

`( x^50   3^9   y^6   x^6)/(3^10   y^8  3^20   y^50   x^15`

3) Group like bases to make it easier to multiply them

`(3^9    x^50 x^6    y^6 )/(3^10 3^20    x^15    y^8 y^50`

4) Multiply like bases
To multiply exponents, you add

`(3^9  x^56  y^6 )/(3^30  x^15  y^58`

5) Reduce by cancelling
To divide exponents, you subtract

`( x^(56-15) )/(3^(30-9)  y^(58-6)`

same as

`( x^41 )/(3^21  y^52` `larr` answer