\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} \chead{\color{black} \Large Algebraic Transformations} \rhead{\ page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ Hidegkuti, Powell, 2009} \textwidth 7.4in \textheight 9.4in \setlength{\headheight}{30pt} \setlength{\parindent}{0in} \begin{document} \begin{enumerate} \item Simplify each of the following expressions. a) \ $\dfrac{a^{2}-9}{a+2}\div \left( 1-\dfrac{5}{a+2}\right) $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c) \ $\dfrac{1-\dfrac{x^{2}}{x^{2}-1}}{2+\dfrac{3x-1}{1-x}}$ \ \ \ where $% \left\vert x\right\vert \not=1$ b) $\ \dfrac{9a-3b}{9a^{2}-b^{2}}\cdot \dfrac{15a+5b}{3}$\ \ \ where \ $% \left\vert 3a\right\vert \not=\left\vert b\right\vert $ \item Find the exact value of each of the following expressions if $x=2,$ \ $% y=\sqrt{3},$ and $z=0.2009$ a) $\ \dfrac{1}{\left( x-y\right) \left( x-z\right) }+\dfrac{1}{\left( z-x\right) \left( z-y\right) }+\dfrac{1}{\left( y-x\right) \left( y-z\right) }$ \ \ \ \ \ \ d) \ $\dfrac{\left( x+y\right) ^{2}-\left( x-y\right) ^{2}}{% 4xy}$ b) $\ \dfrac{1}{x\left( x+z\right) }+\dfrac{1}{z\left( x+z\right) }+\dfrac{1% }{x\left( x-z\right) }+\dfrac{1}{z\left( z-x\right) }$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e) \ $\dfrac{\left( x^{2}-y^{2}-z^{2}-2yz\right) \left( x+y-z\right) }{% \left( x+y+z\right) \left( x^{2}+z^{2}-2xz-y^{2}\right) }$ c) $\ \left( \dfrac{2x^{2}+x}{x^{3}-1}-\dfrac{x+1}{x^{2}+x+1}\right) \left( 1+\dfrac{x+1}{x}-\dfrac{x^{2}+5x}{x^{2}+x}\right) $ \item Simplify each of the following expressions. a) $\ \dfrac{4-a^{2}-2ab-b^{2}}{2+a+b}$ \ where $a+b\not=-2$ \ \ \ \ \ \ \ \ b) $\ \dfrac{a^{2}+b^{2}-c^{2}+2ab}{a^{2}-b^{2}+c^{2}+2ac}$ \ \ where \ $% \left\vert a+c\right\vert \not=\left\vert b\right\vert $ \item Prove that if $a+b+c=0,$ then $a^{3}+a^{2}c+b^{2}c-abc+b^{3}=0$ \item Simplify each of the following expressions. a) \ $\sqrt{12}+\sqrt{75}-\sqrt{147}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g) \ $\sqrt{7-4\sqrt{3}}-\sqrt{7+\sqrt{48}}$ b) $\ \sqrt{28}+\sqrt{7}-\sqrt{63}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h) \ $\sqrt[3]{7+5\sqrt{2}}$ c) $\ \sqrt{\sqrt{41}+4\sqrt{2}}\cdot \sqrt{\sqrt{41}-\sqrt{32}}$ \ \ \ \ \ \ \ \ \ \ i) \ $\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}$ d) $\ \sqrt{5\sqrt{3}+\sqrt{59}}\cdot \sqrt{\sqrt{75}-\sqrt{59}}$ \ \ \ \ \ \ \ \ \ j) \ $\sqrt[3]{10+6\sqrt{3}}+\sqrt[3]{10-6\sqrt{3}}$ e) \ $\left( \sqrt{6+\sqrt{11}}+\sqrt{6-\sqrt{11}}\right) ^{2}$ \ \ \ \ \ \ \ \ \ \ \ k) \ $\sqrt[4]{7-4\sqrt{5}}$ f) \ $\sqrt{7+2\sqrt{6}}-\sqrt{7-2\sqrt{6}}$ \item Simplify each of the following expressions. a) $\ \dfrac{3-\sqrt{5}}{3+\sqrt{5}}+\dfrac{3+\sqrt{5}}{3-\sqrt{5}}$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ b) $\ \left( \dfrac{8}{\sqrt{7}+\sqrt{3}}+% \dfrac{12}{\sqrt{7}-\sqrt{3}}\right) \left( 5\sqrt{7}-\sqrt{3}\right) $ \item Rationalize the denominator in each of the following expressions. a) $\ \dfrac{3}{\sqrt{5}-\sqrt{2}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b) $\ \dfrac{% a}{\sqrt{a}+\sqrt{b}}$ \ \ where $a,b>0$ \ \ \ \ \ \ \ \ \ \ \ \ \ c) $\ \dfrac{\sqrt{7}-\sqrt{2}}{\sqrt{7}+\sqrt{2}}$ \item Which one is greater? a) $\ 2\sqrt{7}$ \ \ or \ \ $\dfrac{1}{\sqrt{7}-\sqrt{6}}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c) $\ \dfrac{7}{5-3\sqrt{2}}$ \ \ \ or \ $\sqrt{72}$ b) $\ \sqrt[4]{4}$ \ \ \ or \ \ \ $\sqrt[5]{5}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ d) \ $2\sqrt{3}$ \ \ or \ \ $\dfrac{1}{\sqrt{3}-\sqrt{2}}$ \item For what values of $k$ \ can we factor out $x+3$ \ from the polynomial $2x^{2}+x+k$? \item Find the exact value of the following expression.% \begin{equation*} \dfrac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2% \sqrt{2}} \end{equation*} \item We divided a line segment into two parts so that the ratio between the shorter and longer part is the same as the ratio between the longer part and the entire line segment. \ If $R$ represents this ratio, find the exact value of the following expression.% \begin{equation*} R^{\left( R^{\left( R^{2}+R^{-1}\right) }+R^{-1}\right) }+R^{-1} \end{equation*} \item Find the integer part in $\left( \sqrt{3}+\sqrt{2}\right) ^{6}$. \item If \ $p,~q,$ \ and $r$ are solutions of the equation \ $% x^{3}-x^{2}+x-2,$ the find the exact value of $p^{3}+q^{3}+r^{3}$. \item Is the number \ $\sqrt[3]{7+4\sqrt{3}}+\sqrt[3]{7-4\sqrt{3}}$ \ a solution of the equation \ $x^{3}-3x-14=0$?\pagebreak \end{enumerate} \begin{center} {\Large Answers\bigskip } \end{center} \begin{enumerate} \item a) \ $a+3$ \ \ \ \ \ \ \ \ b) \ $5$ \ \ \ \ \ \ \ \ \ c) $\ \dfrac{1}{% \left( x+1\right) ^{2}}$ \item a) \ $0$ \ \ \ \ \ \ \ b) \ $0$ \ \ \ \ \ \ \ c) \ $\dfrac{1}{6}$ \ \ \ \ d) \ $1$ \ \ \ \ \ \ \ e) \ $1$ \item a) \ $2-a-b$ \ \ \ \ \ \ \ b) \ $\dfrac{a+b-c}{a-b+c}$ \item Prove that if $a+b+c=0,$ then $a^{3}+a^{2}c+b^{2}c-abc+b^{3}=0$\newline Solution:% \begin{eqnarray*} a^{3}+a^{2}c+b^{2}c-abc+b^{3} &=&a^{3}+b^{3}+a^{2}c+b^{2}c-abc= \\ &=&\left( a+b\right) \left( a^{2}-ab+b^{2}\right) +c\left( a^{2}+b^{2}-ab\right) \\ &=&\left( a+b+c\right) \left( a^{2}-ab+b^{2}\right) \\ &=&0\left( a^{2}-ab+b^{2}\right) =0 \end{eqnarray*} \item a) \ $0$ \ \ \ b) \ $0$ \ \ \ c) \ $3$ \ \ \ d) \ $4$ \ \ \ e) \ $22$ \ \ \ f) \ $2$ \ \ \ g) \ $-2\sqrt{3}$ \ \ \ h) \ $\sqrt{2}+1$ \ \ \ i) \ $4$ \ \ \ j) \ $2$ \ \ \ k) \ $\func{undefined}$ \item a) \ $7$ \ \ \ \ b) \ $172$ \item a) \ $\sqrt{5}+\sqrt{2}$ \ \ \ \ \ \ b) \ $\dfrac{a\left( \sqrt{a}-% \sqrt{b}\right) }{a-b}$ \ \ \ \ \ \ \ c) \ $\dfrac{9-2\sqrt{14}}{5}$ \item the one on the left is greater in all four cases \item $-15$ \item $1$ \item $2$ \item $969$ \item $4$ \item yes \end{enumerate} \bigskip \bigskip \vspace{0.4in} \vspace{0.4in} \vspace{1in} %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}