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%TCIDATA{Created=Wednesday, September 21, 2005 19:23:38}
%TCIDATA{LastRevised=Sunday, December 04, 2005 23:39:10}
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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
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\begin{document}


\begin{center}
{\Large Exponent}
\end{center}

Simplify.

\begin{enumerate}
\item $\left( \dfrac{16x^{4}y^{\frac{2}{3}}}{36x^{2}y^{2}}\right) ^{\frac{3}{%
2}}=$%
\begin{eqnarray*}
\left( \dfrac{16x^{4}y^{\frac{2}{3}}}{36x^{2}y^{2}}\right) ^{\frac{3}{2}} &=&%
\dfrac{16^{\frac{3}{2}}\cdot x^{4\cdot \frac{3}{2}}\cdot y^{\frac{2}{3}\cdot 
\frac{3}{2}}}{36^{\frac{3}{2}}\cdot x^{2\cdot \frac{3}{2}}\cdot y^{6\cdot 
\frac{3}{2}}}=\dfrac{64\cdot x^{6}\cdot y^{\frac{2}{3}\cdot \frac{3}{2}}}{%
216\cdot x^{3}\cdot y^{\frac{18}{2}}}=\dfrac{64\cdot x^{6}\cdot y}{216\cdot
x^{3}\cdot y^{9}}= \\
&=&\dfrac{64\cdot x^{3}}{216\cdot y^{8}}=\dfrac{8x^{3}}{27y^{8}}
\end{eqnarray*}

\item $\allowbreak \left( \dfrac{9x^{6}y^{16}}{36x^{2}y^{8}}\right) ^{\frac{1%
}{2}}=$%
\[
\left( \dfrac{9x^{6}y^{16}}{36x^{2}y^{8}}\right) ^{\frac{1}{2}}=\dfrac{9^{%
\frac{1}{2}}\cdot x^{6\cdot \frac{1}{2}}\cdot y^{16\cdot \frac{1}{2}}}{36^{%
\frac{1}{2}}\cdot x^{2\cdot \frac{1}{2}}\cdot y^{8\cdot \frac{1}{2}}}=\dfrac{%
3\cdot x^{3}\cdot y^{8}}{6\cdot x^{1}\cdot y^{4}}=\dfrac{1\cdot x^{3-1}\cdot
y^{8-4}}{3}=\dfrac{x^{2}y^{4}}{3} 
\]

\item $\left( \dfrac{1x^{2}y^{12}}{4x^{8}y^{6}}\right) ^{\frac{1}{2}}=$%
\begin{eqnarray*}
\left( \dfrac{1x^{2}y^{12}}{4x^{8}y^{6}}\right) ^{\frac{1}{2}} &=&\dfrac{1^{%
\frac{1}{2}}\cdot x^{2\cdot \frac{1}{2}}\cdot y^{12\cdot \frac{1}{2}}}{4^{%
\frac{1}{2}}\cdot x^{8\cdot \frac{1}{2}}\cdot y^{6\cdot \frac{1}{2}}}=\dfrac{%
1\cdot x^{1}\cdot y^{6}}{2\cdot x^{4}\cdot y^{3}}=\dfrac{1\cdot x^{1-4}\cdot
y^{6-3}}{2}=\dfrac{1\cdot x^{-3}\cdot y^{3}}{2}= \\
&=&\dfrac{1y^{3}}{2x^{3}}\text{ or }\dfrac{y^{3}}{2x^{3}}
\end{eqnarray*}

\item $\left( \dfrac{27x^{3}y^{12}}{8x^{6}y^{9}}\right) ^{\frac{1}{3}}=$%
\begin{eqnarray*}
\left( \dfrac{27x^{3}y^{12}}{8x^{6}y^{9}}\right) ^{\frac{1}{3}} &=&\dfrac{%
27^{\frac{1}{3}}\cdot x^{3\cdot \frac{1}{3}}\cdot y^{12\cdot \frac{1}{3}}}{%
8^{\frac{1}{3}}\cdot x^{6\cdot \frac{1}{3}}\cdot y^{9\cdot \frac{1}{3}}}=%
\dfrac{3\cdot x^{1}\cdot y^{4}}{2\cdot x^{2}\cdot y^{3}}=\dfrac{3\cdot
x^{1-2}\cdot y^{4-3}}{2}=\dfrac{3\cdot x^{-1}\cdot y^{1}}{2}= \\
&=&\dfrac{3y^{1}}{2x^{1}}=\dfrac{3y}{2x}
\end{eqnarray*}

\item $\left( \dfrac{64x^{15}y^{6}}{8x^{9}y^{3}}\right) ^{\frac{1}{3}}=$%
\begin{eqnarray*}
\left( \dfrac{64x^{15}y^{6}}{8x^{9}y^{3}}\right) ^{\frac{1}{3}} &=&\dfrac{%
64^{\frac{1}{3}}\cdot x^{15\cdot \frac{1}{3}}\cdot y^{6\cdot \frac{1}{3}}}{%
8^{\frac{1}{3}}\cdot x^{9\cdot \frac{1}{3}}\cdot y^{3\cdot \frac{1}{3}}}=%
\dfrac{4\cdot x^{5}\cdot y^{2}}{2\cdot x^{3}\cdot y^{1}}=\dfrac{2\cdot
x^{5-3}\cdot y^{2-1}}{1}=2\cdot x^{2}\cdot y^{1}= \\
&=&2x^{2}y^{1}
\end{eqnarray*}

\item $x^{\frac{2}{9}}\left( x^{-\frac{2}{9}}+x^{\frac{7}{9}}\right) =$%
\[
x^{\frac{2}{9}}\left( x^{-\frac{2}{9}}+x^{\frac{7}{9}}\right) =x^{\frac{2}{9}%
}\cdot x^{-\frac{2}{9}}+x^{\frac{2}{9}}\cdot x^{\frac{7}{9}}=x^{\frac{2}{9}-%
\frac{2}{9}}+x^{\frac{2}{9}+\frac{7}{9}}=x^{0}+x^{\frac{9}{9}}=1+x^{1}=1+x
\]

\item $x^{\frac{2}{3}}\left( x^{\frac{4}{3}}+x^{-\frac{2}{3}}\right) =$%
\[
x^{\frac{2}{3}}\left( x^{\frac{4}{3}}+x^{-\frac{2}{3}}\right) =x^{\frac{2}{3}%
}\cdot x^{\frac{4}{3}}+x^{\frac{2}{3}}\cdot x^{-\frac{2}{3}}=x^{\frac{2}{3}+%
\frac{4}{3}}+x^{\frac{2}{3}-\frac{2}{3}}=x^{\frac{6}{3}%
}+x^{0}=x^{3}+1=x^{3}+1
\]

\item $x^{\frac{1}{2}}\left( x^{\frac{4}{3}}+x^{\frac{5}{3}}\right) =$%
\[
x^{\frac{1}{2}}\left( x^{\frac{4}{3}}+x^{\frac{5}{3}}\right) =x^{\frac{1}{2}%
}\cdot x^{\frac{4}{3}}+x^{\frac{1}{2}}\cdot x^{\frac{5}{3}}=x^{\frac{1}{2}+%
\frac{4}{3}}+x^{\frac{1}{2}+\frac{5}{3}}=x^{\frac{3}{6}+\frac{8}{6}}+x^{%
\frac{3}{6}+\frac{10}{6}}=x^{\frac{11}{6}}+x^{\frac{13}{6}} 
\]

\item $x^{\frac{1}{3}}\left( x^{\frac{2}{3}}+x^{\frac{7}{9}}\right) =$%
\[
x^{\frac{1}{3}}\left( x^{\frac{2}{3}}+x^{\frac{7}{9}}\right) =x^{\frac{1}{3}%
}\cdot x^{\frac{2}{3}}+x^{\frac{1}{3}}\cdot x^{\frac{7}{9}}=x^{\frac{1}{3}+%
\frac{2}{3}}+x^{\frac{1}{3}+\frac{7}{9}}=x^{\frac{3}{3}}+x^{\frac{3}{9}+%
\frac{7}{9}}=x+x^{\frac{10}{9}} 
\]

\item $x^{\frac{5}{8}}\left( x^{\frac{3}{8}}+x^{\frac{7}{8}}\right) =$%
\[
x^{\frac{5}{8}}\left( x^{\frac{3}{8}}+x^{\frac{7}{8}}\right) =x^{\frac{5}{8}%
}\cdot \left( x^{\frac{5}{8}}\cdot x^{\frac{3}{8}}+x^{\frac{5}{8}}\cdot x^{%
\frac{7}{8}}\right) =x^{\frac{3}{8}+\frac{5}{8}}+x^{\frac{7}{8}+\frac{5}{8}%
}=x^{\frac{8}{8}}+x^{\frac{12}{8}}=x+x^{\frac{3}{2}} 
\]
\end{enumerate}

\end{document}