Skip to main content
\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

AppendixCSome Useful Rules of Algebra

  • For positive integers \(m\) and \(n\), \(\sqrt[n]{x^m}=x^{\sfrac{m}{n}}\). This is universally true when \(x\) is positive. When \(x\) is negative, things are more complicated. It boils down to a choice. Some computer algebra systems choose to define an expression like \((-8)^{\sfrac{1}{3}}\) as a certain negative real number (\(-2\)). Others choose to define an expression like \((-8)^{\sfrac{1}{3}}\) as a complex number in the upper half plane of the complex plane (\(1+i\sqrt{3}\)). Because of the ambiguity, some choose to simply declare expressions like \((-8)^{\sfrac{1}{3}}\), with a negative base, to be undefined. The point is, if your computer tells you that say, \((-8)^{\sfrac{1}{3}}\) is undefined, you should realize that you may be expected to interpret \((-8)^{\sfrac{1}{3}}\) as \(-2\).

  • For real numbers \(k\) and \(n\), and \(x\neq0\), \(\frac{k}{x^n}=kx^{-n}\).

  • For a real number \(k\neq0\), \(\frac{\fe{f}{x}}{k}=\frac{1}{k}\fe{f}{x}\).

  • For positive real numbers \(A\) and \(B\) and all real numbers \(n\),

    • \(\fe{\ln}{AB}=\fe{\ln}{A}+\fe{\ln}{B}\)

    • \(\fe{\ln}{\frac{A}{B}}=\fe{\ln}{A}-\fe{\ln}{B}\)

    • \(\fe{\ln}{A^n}=n\fe{\ln}{A}\)