\hfill \hfill } %} \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}{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}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.5in} \pagestyle{fancy} \lhead{\color{blue} \large Practice} \chead{\color{black} \Large Integer Exponents} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2011} \rfoot{\small Last revised: November 3, 2011} \textwidth 7.5in \textheight 9.5in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} Simplify each of the following. \ Assume that all variables represent positive numbers. \ \ Present your answer without negative exponents.\medskip\ %TCIMACRO{\TeXButton{3col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion \medskip \begin{enumerate} \item $3^{-2}$\medskip \item $\dfrac{1}{2^{-3}}$\medskip \item $m^{-4}$\medskip \item $\dfrac{1}{x^{-5}}$\medskip \item $a^{8}\cdot a^{-1}$\medskip \item $p^{3}\left( p^{-7}\right) p^{8}$\medskip \item $\dfrac{x^{-4}}{x^{-9}}$\medskip \item $\dfrac{50a^{12}}{10a^{-3}}$\medskip \item $\dfrac{t^{-3}}{t^{4}}$\medskip \item $x^{0}$\medskip \item $-x^{0}$\medskip \item $\left( -x\right) ^{0}$\medskip \item $\left( b^{-5}\right) \left( b^{2}\right) \left( b^{-1}\right) $% \medskip \item $\dfrac{1}{\left( b^{-5}\right) \left( b^{2}\right) \left( b^{-1}\right) }$\medskip \item $\dfrac{m^{-2}}{m^{-5}}$\medskip \item $\dfrac{x^{3}y^{-5}}{z^{-4}}$\medskip \item $\dfrac{18q^{3}}{6q^{-3}}$\medskip \item $\left( \dfrac{2}{3}\right) ^{-3}$\medskip \item $2y^{-3}$\medskip \item $\left( 2y\right) ^{-3}$\medskip \item $\left( -\dfrac{3}{5}\right) ^{-2}$\medskip \item $\dfrac{a^{3}b^{-5}}{a^{-2}b^{3}}$\medskip \item $\left( 3m^{3}\right) ^{-2}$\medskip \item $\left( -2ab^{-3}\right) ^{-3}$\medskip \item $\dfrac{\left( k^{3}\right) ^{-3}}{\left( k^{-5}\right) ^{2}}$\medskip \item $\left( \dfrac{2a^{-3}b^{5}}{-3a^{3}b^{-2}}\right) ^{-2}\left( a^{3}b^{-5}\right) ^{-3}$\medskip \item $\left( -2a^{-3}\right) \left( -2a^{-2}b\right) ^{-4}$\medskip \item $\dfrac{\left( -3p^{3}q^{5}\right) ^{2}}{\left( 2q^{0}p^{3}\right) ^{-1}}$\medskip \item $\left( \dfrac{2a^{-2}b^{3}}{-2^{2}\left( a^{-1}b\right) ^{-3}}\right) ^{-2}$\medskip \item $\left( -\dfrac{x^{3}y^{0}x^{-5}}{y^{-3}}\right) ^{-2}$\medskip \item $\left( -\dfrac{x^{3}y^{7}x^{-5}}{y^{-3}}\right) ^{0}$\medskip \item $\dfrac{x^{-1}+y^{-1}}{x^{-2}-y^{-2}}$\medskip \item $\dfrac{\left( -2a^{-2}\right) ^{-2}b^{3}a^{0}\left( -aba^{-2}b^{-2}\right) ^{-3}}{2a^{2}\left( -2a^{-2}b\right) ^{-2}ab^{0}}$% \medskip \item $\left( \dfrac{-a^{2}\left( b^{-1}a\right) ^{-5}}{b^{7}\left( -ab^{2}\right) ^{-3}}\right) ^{-2}$\medskip \item $\dfrac{\left( x^{-2}\right) ^{-2}y^{3}x^{0}\left( -2yx^{0}y^{-2}x^{-2}\right) ^{0}}{yx^{5}\left( y^{-2}x\right) ^{-3}\left( 2x^{-1}yx^{3}\right) ^{-1}}$\ \ \end{enumerate} \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \ \ \pagebreak \bigskip \begin{center} {\LARGE Answers\bigskip } \end{center} %TCIMACRO{\TeXButton{5col begin}{\begin{multicols}{5}}}% %BeginExpansion \begin{multicols}{5}% %EndExpansion \bigskip \begin{enumerate} \item $\dfrac{1}{9}$\medskip \item $8$\medskip \item $\dfrac{1}{m^{4}}$\medskip \item $x^{5}$\medskip \item $a^{7}$\medskip \item $p^{4}$\medskip \item $x^{5}$\medskip \item $5a^{15}$\medskip \item $\dfrac{1}{t^{7}}$\medskip \item $1$\medskip \item $-1$\medskip \item $1$\medskip \item $\dfrac{1}{b^{4}}$\medskip \item $b^{4}$\medskip \item $m^{3}$\medskip \item $\dfrac{x^{3}z^{4}}{y^{5}}$\medskip \item $3q^{6}$\medskip \item $\dfrac{27}{8}$\medskip \item $\dfrac{2}{y^{3}}$\medskip \item $\dfrac{1}{8y^{3}}$\medskip \item $\dfrac{25}{9}$\medskip \item $\dfrac{a^{5}}{b^{8}}$\medskip \item $\dfrac{1}{9m^{6}}$\medskip \item $-\dfrac{b^{9}}{8a^{3}}$\medskip \item $k$\medskip \item $\dfrac{9}{4}a^{3}b$\medskip \item $-\dfrac{a^{5}}{8b^{4}}$\medskip \item $18p^{9}q^{10}$\medskip \item $\dfrac{4a^{10}}{b^{12}}$\medskip \item $\dfrac{x^{4}}{y^{6}}$\medskip \item $1$\medskip \item $\dfrac{xy}{y-x}$\medskip \item $-\dfrac{b^{8}}{2}$\medskip \item $\dfrac{1}{b^{8}}$\medskip \item $\dfrac{2x^{4}}{y^{3}}$\medskip \end{enumerate} %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \vspace{0.6in} \vspace{0.7in} \vspace{3in} \bigskip \bigskip \bigskip \bigskip \href{http://www.teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html% }{For more documents like this, visit our page at\ http://www.teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}