[tex] \bf ~~~~~~~~~~~~\textit{negative exponents}
\\\\
a^{-n} \implies \cfrac{1}{a^n}
\qquad \qquad
\cfrac{1}{a^n}\implies a^{-n}
\qquad \qquad
a^n\implies \cfrac{1}{a^{-n}}
\\\\
------------------------------- [/tex]
[tex] \bf \cfrac{(4x^2y^3z^5)^3}{(2x^5y^2z^6)^3}\implies \stackrel{\textit{distributing the exponent}}{\cfrac{(4^3x^{2\cdot 3}y^{3\cdot 3}z^{5\cdot 3})}{(2^3x^{5\cdot 3}y^{2\cdot 3}z^{6\cdot 3})}}\implies \cfrac{64x^6y^9z^{15}}{8x^{15}y^6z^{18}}
\\\\\\
\cfrac{64}{8}\cdot \cfrac{x^6y^9z^{15}}{x^{15}y^6z^{18}}\implies \cfrac{8}{1}\cdot \cfrac{y^9y^{-6}}{x^{15}x^{-6}z^{18}z^{-15}}\implies 8\cdot \cfrac{y^{9-6}}{x^{15-6}z^{18-15}}
\\\\\\
\cfrac{8y^3}{x^9z^3} [/tex]
(4x^2y^3z^5)^3
------------------- Your going to multiply everything by 3
(2x^5y^2z^6)^3
12x^6y^9z^15
--------------------
6x^15y^6z^18 Then you'll cancel things out. And remember you cant cancel exponents and variables that aren't the same.
So 12/6=2 x^15-x^6=x^9 y^9-y^6=y^3 z^18-z^15^3
If you cancel all the numbers to the top you get negative so I did it that you come out all positive.
2y^3
----------
x^9z^3
If you want it all at the top it'll be 2x^-9y^3z^-3
