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%TCIDATA{<META NAME="Title" CONTENT="Negative Exponents">}
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\chead{\color{black} \Large Negative Exponents}
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\lfoot{\small   \copyright $\;$ copyright  Hidegkuti,  Powell,  2011}
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\begin{document}


Simplify each of the following. \ Assume that all variables represent
positive numbers. \ \ Present your answer without negative exponents.

\begin{enumerate}
\item $3^{-2}=~~%
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\dfrac{1}{9}$%
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\item $\dfrac{1}{2^{-3}}=\allowbreak 8$

\item $m^{-4}=\dfrac{1}{m^{4}}$

\item $\dfrac{1}{x^{-5}}=~~%
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x^{5}$%
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\item $a^{8}\cdot a^{-1}=~~%
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a^{7}$%
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\item $p^{3}\left( p^{-7}\right) p^{8}=~~%
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p^{4}$%
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\item $\dfrac{x^{-4}}{x^{-9}}=~~%
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x^{5}$%
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\item $\dfrac{50a^{12}}{10a^{-3}}=~~%
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5a^{15}$%
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\item $\dfrac{t^{-3}}{t^{4}}=~~%
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\dfrac{1}{t^{7}}$%
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\item $x^{0}=1$

\item $-x^{0}=-1$

\item $\left( -x\right) ^{0}=\allowbreak 1$

\item $\left( b^{-5}\right) \left( b^{2}\right) \left( b^{-1}\right) =~~%
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\dfrac{1}{b^{4}}$%
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\item $\dfrac{1}{\left( b^{-5}\right) \left( b^{2}\right) \left(
b^{-1}\right) }=\allowbreak b^{4}$

\item $\dfrac{m^{-2}}{m^{-5}}=~~%
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m^{3}$%
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\item $\dfrac{x^{3}y^{-5}}{z^{-4}}=\dfrac{x^{3}z^{4}}{y^{5}}$

\item $\dfrac{18q^{3}}{6q^{-3}}=~~%
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3q^{6}$%
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\item $\left( \dfrac{2}{3}\right) ^{-3}=\dfrac{27}{8}$

\item $2y^{-3}=~~%
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\dfrac{2}{y^{3}}$%
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\item $\left( 2y\right) ^{-3}=~~%
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\dfrac{1}{8y^{3}}$%
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\item $\left( -\dfrac{3}{5}\right) ^{-2}=\dfrac{25}{9}$

\item $\dfrac{a^{3}b^{-5}}{a^{-2}b^{3}}=~~%
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\dfrac{a^{5}}{b^{8}}$%
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\item $\left( 3m^{3}\right) ^{-2}=~~%
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\dfrac{1}{9m^{6}}$%
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\item $\left( -2ab^{-3}\right) ^{-3}=-\dfrac{b^{9}}{8a^{3}}$

\item $\dfrac{\left( k^{3}\right) ^{-3}}{\left( k^{-5}\right) ^{2}}=~~%
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k$%
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\item $\left( \dfrac{2a^{-3}b^{5}}{-3a^{3}b^{-2}}\right) ^{-2}\left(
a^{3}b^{-5}\right) ^{-3}=\dfrac{9}{4}a^{3}b$

\item $\left( -2a^{-3}\right) \left( -2a^{-2}b\right) ^{-4}=-\dfrac{a^{5}}{%
8b^{4}}$

\item $\dfrac{\left( -3p^{3}q^{5}\right) ^{2}}{\left( 2q^{0}p^{3}\right)
^{-1}}=18p^{9}q^{10}$

\item $\left( \dfrac{2a^{-2}b^{3}}{-2^{2}\left( a^{-1}b\right) ^{-3}}\right)
^{-2}=~~%
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\dfrac{4a^{10}}{b^{12}}$%
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\item $\left( -\dfrac{x^{3}y^{0}x^{-5}}{y^{-3}}\right) ^{-2}=\dfrac{x^{4}}{%
y^{6}}$

\item $\left( -\dfrac{x^{3}y^{7}x^{-5}}{y^{-3}}\right) ^{0}=1$

\item $\dfrac{x^{-1}+y^{-1}}{x^{-2}-y^{-2}}=~~%
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\dfrac{xy}{y-x}$%
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\item $\dfrac{\left( -2a^{-2}\right) ^{-2}b^{3}a^{0}\left(
-aba^{-2}b^{-2}\right) ^{-3}}{2a^{2}\left( -2a^{-2}b\right) ^{-2}ab^{0}}=-%
\dfrac{b^{8}}{2}$

\item $\left( \dfrac{-a^{2}\left( b^{-1}a\right) ^{-5}}{b^{7}\left(
-ab^{2}\right) ^{-3}}\right) ^{-2}=\dfrac{1}{b^{8}}$

\item $\dfrac{\left( x^{-2}\right) ^{-2}y^{3}x^{0}\left(
-2yx^{0}y^{-2}x^{-2}\right) ^{0}}{yx^{5}\left( y^{-2}x\right) ^{-3}\left(
2x^{-1}yx^{3}\right) ^{-1}}=\dfrac{2x^{4}}{y^{3}}$
\end{enumerate}

\end{document}