[tex]\bf \left.\qquad \qquad \right.\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}}}}\\\\
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\left( \cfrac{32x^{18}y^{10}}{16x^9y^{20}} \right)^2\impliedby \textit{first off, let's distribute the exponent}
\\\\\\
[/tex]
[tex]\bf \left( \cfrac{32^2x^{2\cdot 18}y^{2\cdot 10}}{16^2x^{2\cdot 9}y^{2\cdot 20}} \right)\implies \cfrac{32^2x^{36}y^{20}}{16^2x^{18}y^{40}}\implies
\cfrac{32^2x^{36}x^{-18}}{16^2y^{-20}y^{40}}
\\\\\\[/tex]
[tex]\bf \cfrac{32^2x^{36-18}}{16^2y^{-20+40}}
\implies
\cfrac{32^2x^{18}}{16^2y^{20}}\implies \cfrac{32\cdot 32}{16\cdot 16}\cdot \cfrac{x^{18}}{y^{20}}\implies \cfrac{32}{16}\cdot \cfrac{32}{16}\cdot \cfrac{x^{18}}{y^{20}}
\\\\\\
\cfrac{2}{1}\cdot \cfrac{2}{1}\cdot \cfrac{x^{18}}{y^{20}}\implies \cfrac{4x^{18}}{y^{20}}[/tex]
