Question 169632
Note: 


{{{(x^(-2)y^(8)z)^(-5)=1/(x^(-2)y^(8)z)^5}}}



and 



{{{1/(x^(-4)y^(9)z)^(-4)=(x^(-4)y^(9)z)^(4)}}}



In other words, we just flip the fractions



So 


{{{((x^(-2)y^(8)z)^(-5))/((x^(-4)y^(9)z)^(-4))}}} becomes




{{{((x^(-4)y^(9)z)^(4))/((x^(-2)y^(8)z)^5)}}}



{{{(x^(-4*4)y^(9*4)z^(1*4))/(x^(-2*5)y^(8*5)z^(1*5))}}} Distribute the exponents.



{{{(x^(-16)y^(36)z^(4))/(x^(-10)y^(40)z^(5))}}} Multiply the exponents.



{{{x^(-16--10)y^(36-40)z^(4-5)}}} Subtract the exponents



For example, {{{(x^(-16))/(x^(-10))=x^(-16--10)}}}



{{{x^(-16+10)y^(36-40)z^(4-5)}}} Simplify



{{{x^(-6)y^(-4)z^(-1)}}} Combine the exponents.



{{{(1/x^(6))(1/y^(4))(1/z^(1))}}} Flip the variables with negative exponents.



{{{1/(x^6y^4z^1)}}} Combine the fractions.



{{{1/(x^6y^4z)}}} Simplify



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Answer:



So {{{((x^(-2)y^(8)z)^(-5))/((x^(-4)y^(9)z)^(-4))}}} simplifies to {{{1/(x^6y^4z)}}}



In other words, {{{((x^(-2)y^(8)z)^(-5))/((x^(-4)y^(9)z)^(-4))=1/(x^6y^4z)}}} where no variables can be equal to zero.