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\Huge{Exponent Properties}
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\noindent
There are `five' properties of exponents that you need to know well to work with algebraic expressions and equations.   The `zeroth' property is:



\vskip .2in
\large \noindent Property 0) Let $b$ be a real number and $m$ a positive integer, then $$b^m=b\cdot b \cdot b \cdot  \ldots  \cdot b$$
\small where $b$ appears $m$ times on the right.
\vskip .2in
\normalsize For example, we can write out $(-2)^4$ with no exponents as $(-2)\cdot (-2) \cdot (-2) \cdot (-2)$ or $(4)^3$ as $4\cdot 4\cdot 4$.  We will often rely on the order of operations and drop parenthesis and just write $4^3$ instead.  

Warning: the order of operations is such that exponents \emph{only} effect the object they `hover' behind.  For example,$-2^4=-2\cdot 2\cdot 2\cdot 2$, since the exponent is only effecting the $2$ and not the negative sign.

There are some special cases of the above definition (that not all mathematicians agree on).  Mainly:

\hspace{1in} $0^m=0$ \hspace{.5in} $1^m=1$ \hspace{.5in} $b^0=1$ \hspace{.5in} $0^0$ is undefined

\begin{enumerate}
\item Consider $x^2\cdot x^5$. 
\begin{enumerate}
\item Write out $x^2\cdot x^5$ with no exponents (like what was done in the above paragraph).
\vfill
\item Write $x^2\cdot x^5$ as $x$ raised to a single power.
\vfill
\end{enumerate}
%\item Consider $y^3\cdot y^3$.  
%\begin{enumerate}
%\item Write out $y^3\cdot y^3$with no exponents.
%\vfill
%\item Write $y^3\cdot y^3$ as $y$ raised to a single power.
%\vfill
%\end{enumerate}
\item Use your work above the finish the following first property:
\end{enumerate}

\vskip .2in
\large
\noindent Property 1) Let $b$ be a real number, and let $m$ and $n$ be positive integers, then $$b^m\cdot b^n=  $$

\vskip .2in
\normalsize 


\newpage
\begin{enumerate}
\item Write out $\square ^4$ with no exponents (like you did on the first page).
\vskip .7in
\item Write an $(x^3)^4$ using only multiplication and instances of $x^3$.  In other words, do the same problem as number one but instead of using a square, use the symbol $x^3$.
\vskip 1in
\item Now expand the above with Property 0 to write out $(x^3)^4$ with no exponents.
\vskip 1in
\item Repeat the steps above to write out $(y^5)^2$ with no exponents.
\vskip 1in
\item Use your work above to finish the following:
\end{enumerate}



\vskip .2in
\large
\noindent Property 2) Let $b$ be a real number, and let $m$ and $n$ be positive integers, then $$(b^m)^n=  $$

\vskip .2in
\normalsize 

\newpage
\begin{enumerate} 
\item Let $x$ and $y$ be real numbers and $m$ be positive integers.  Is there a relationship between $(x\cdot y)^m$ and $x^m\cdot y^m$?  If so, state it and explain why it is true.
\end{enumerate}
\vspace{1in}

By answering the previous question you can now state:

\vskip .2in
\large
\noindent Property 3) Let $a$ and $b$ be real numbers, and let $m$ be a positive integer, then $$(a\cdot b)^m=  $$

\vskip .2in
\normalsize You can verify these properties on page 14 of your text.  I suggest you \emph{do not} memorize them as most people mix them up.  Rather, work through examples like 2 on page 1 and 4 on page 2 whenever you realize you need to work with exponents.


The remaining two properties are:

\vskip .2in
\large
\noindent Property 4) Let $b$ be a real number and let $m$ and $n$ be positive integers, then $$\displaystyle{\frac{b^m}{b^n}=b^{m-n}}.$$
\vskip .2in \normalsize and

\vskip .2in
\large 
\noindent Property 5) Let $a$ and $b$ be a real numbers and let $m$ be a positive integer, then $$\displaystyle{\frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m}.$$

 \normalsize
\begin{enumerate}
\item Recall that $x^{-2}=\frac{1}{x^2}$.  Make sure Property 4 is consistent with this observation by simplifying $$\displaystyle{\frac{x^6}{x^8}}$$
\vfill


\end{enumerate}


\newpage
\noindent All the rules can be found in Appendix 1 on pages 788 \& 189.  Now, some practice...

\begin{enumerate}

\item Simplify $(-4x^2y^3)(7x^3y^{-1})$.
\vskip 1.0in
\item Simplify $\displaystyle{\left(\frac{x^5}{2y^{-3}}\right)^{-3}}$. Hint: use \emph{lots} of room and make sure your exponents don't fall!
\vskip 2in



\item Solve for $x$ (real or complex) given:\\
 $3x^{-2}+7=0$ \hspace{2in} $(x+1)^{-1}+\frac{1}{2}=(x+3)^{-1}$
\vfill


\end{enumerate}
\vfill

\small Notice that you can verify your answers to 1-2 by looking at Example 7 on page 790. Number four comes up a great deal in calculus word problems.


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