# AppendixCSome Useful Rules of Algebra¶ permalink

• For positive integers $m$ and $n\text{,}$ $\sqrt[n]{x^m}=x^{\sfrac{m}{n}}\text{.}$ 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 (namely, $-2$). 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 $(-8)^{\sfrac{1}{3}}\text{,}$ 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 $-2\text{.}$

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

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

• For positive real numbers $A$ and $B$ and all real numbers $n\text{,}$

• $\fe{\ln}{AB}=\fe{\ln}{A}+\fe{\ln}{B}$

• $\fe{\ln}{\frac{A}{B}}=\fe{\ln}{A}-\fe{\ln}{B}$

• $\fe{\ln}{A^n}=n\fe{\ln}{A}$