\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=0.6in,right=0.7in,top=0.7in,bottom=0.7in} \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 7.4in \textheight 9.4in \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 \medskip \qquad 1.) $\ 3^{-2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ 13.) $\ \left( b^{-5}\right) \left( b^{2}\right) \left( b^{-1}\right) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.) \ $\dfrac{\left( k^{3}\right) ^{-3}}{\left( k^{-5}\right) ^{2}}$\ \qquad 2.) $\ \dfrac{1}{2^{-3}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ 14.) $\ \dfrac{1}{\left( b^{-5}\right) \left( b^{2}\right) \left( b^{-1}\right) }$\ \ \ \ \ \ \ \ \ \ \ \ \ 26.) \ $\left( \dfrac{2a^{-3}b^{5}% }{-3a^{3}b^{-2}}\right) ^{-2}\left( a^{3}b^{-5}\right) ^{-3}$ \qquad 3.) $\ m^{-4}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ 15.) \ $\dfrac{m^{-2}}{m^{-5}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 27.) \ $\left( -2a^{-3}\right) \left( -2a^{-2}b\right) ^{-4}$\ \qquad 4.) $\ \dfrac{1}{x^{-5}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ 16.) \ $\dfrac{x^{3}y^{-5}}{z^{-4}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 28.) \ $\dfrac{\left( -3p^{3}q^{5}\right) ^{2}}{\left( 2q^{0}p^{3}\right) ^{-1}}$\ \qquad 5.) $\ a^{8}\cdot a^{-1}$ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ 17.) \ $\dfrac{18q^{3}}{6q^{-3}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 29.) \ $\left( \dfrac{2a^{-2}b^{3}}{-2^{2}\left( a^{-1}b\right) ^{-3}}\right) ^{-2}$\ \ \ \qquad 6.) $\ p^{3}\left( p^{-7}\right) p^{8}$ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ 18.) \ $\left( \dfrac{2}{3}\right) ^{-3}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30.) \ $\left( -\dfrac{x^{3}y^{0}x^{-5}}{y^{-3}}\right) ^{-2} $ \qquad 7.) $\ \dfrac{x^{-4}}{x^{-9}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ 19.) \ $2y^{-3}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 31.) \ $\left( -\dfrac{x^{3}y^{7}x^{-5}}{y^{-3}}\right) ^{0}$ \qquad 8.) $\ \dfrac{50a^{12}}{10a^{-3}}$ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ \ 20.) \ $\left( 2y\right) ^{-3}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 32.) \ $\dfrac{x^{-1}+y^{-1}}{x^{-2}-y^{-2}}$\ \qquad 9.) $\ \dfrac{t^{-3}}{t^{4}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ \ 21.) \ $\left( -\dfrac{3}{5}\right) ^{-2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33.) \ $\dfrac{\left( -2a^{-2}\right) ^{-2}b^{3}a^{0}\left( -aba^{-2}b^{-2}\right) ^{-3}}{2a^{2}\left( -2a^{-2}b\right) ^{-2}ab^{0}}$ \qquad 10.) $\ x^{0}$ \ \ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ \ \ 22.) \ $\dfrac{a^{3}b^{-5}}{a^{-2}b^{3}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 34.) \ $\left( \dfrac{-a^{2}\left( b^{-1}a\right) ^{-5}}{% b^{7}\left( -ab^{2}\right) ^{-3}}\right) ^{-2}$ \qquad 11.) $\ -x^{0}$ \ \ \ \ \ \ \ \ \ \ \medskip\ \ \ \ \ \ \ \ \ 23.) \ $% \left( 3m^{3}\right) ^{-2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35.) \ $\dfrac{\left( x^{-2}\right) ^{-2}y^{3}x^{0}\left( -2yx^{0}y^{-2}x^{-2}\right) ^{0}}{yx^{5}\left( y^{-2}x\right) ^{-3}\left( 2x^{-1}yx^{3}\right) ^{-1}}$ \qquad 12.) $\ \left( -x\right) ^{0}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 24.) \ $\left( -2ab^{-3}\right) ^{-3}$ \ \ \ \ \ \ \pagebreak \bigskip \begin{center} {\Large Answers} \bigskip \end{center} 1.) $\ \dfrac{1}{9}$ \ \ \ \ 2.) $\ 8$ \ \ \ \ 3.) $\ \dfrac{1}{m^{4}}$ \ \ \ \ 4.) $\ x^{5}$ \ \ \ \ 5.) $\ a^{7}$ \ \ \ \ 6.) $\ p^{4}$ \ \ \ \ 7.) $\ x^{5}$ \ \ \ \ 8.) $\ 5a^{15}$ \ \ \ \ 9.) $\ \dfrac{1}{t^{7}}$ \ \ \ 10.) $% \ 1$\bigskip 11.) $\ -1$ \ \ \ \ 12.) $\ 1$ \ \ \ 13.) $\ \dfrac{1}{b^{4}}$ \ \ \ 14.) \ $% b^{4}$ \ \ \ 15.) $\ m^{3}$ \ \ \ 16.) $\ \dfrac{x^{3}z^{4}}{y^{5}}$ \ \ \ 17.) $\ 3q^{6}$ \ \ \ 18.) $\ \dfrac{27}{8}$ \ \ \ 19.) $\ \dfrac{2}{y^{3}}$ \bigskip 20.) $\ \dfrac{1}{8y^{3}}$ \ \ \ \ 21.) $\ \dfrac{25}{9}$ \ \ \ \ 22.) $\ \dfrac{a^{5}}{b^{8}}$ \ \ \ \ 23.) $\dfrac{1}{9m^{6}}$ \ \ \ \ 24.) $\ -% \dfrac{b^{9}}{8a^{3}}$ \ \ \ \ 25.) $\ k$ \ \ \ \ 26.) $\ \dfrac{9}{4}a^{3}b$ \ \ \ \ \ 27.) $\ -\dfrac{a^{5}}{8b^{4}}$\bigskip 28.) $\ 18p^{9}q^{10}$ \ \ \ \ 29.) $\ \dfrac{4a^{10}}{b^{12}}$ \ \ \ \ 30.) $\ \dfrac{x^{4}}{y^{6}}$ \ \ \ \ 31.) $\ 1$ \ \ \ \ 32.) $\ \dfrac{xy}{y-x}$ \ \ \ \ 33.) $\ -\dfrac{b^{8}}{2}$ \ \ \ \ 34.) $\ \dfrac{1}{b^{8}}$ \ \ \ \ 35.) $\ \dfrac{2x^{4}}{y^{3}}$ \end{document}