Laws of Exponents

[latex]\begin{array}{ccccccccccccc}\hfill {x}^{m}{x}^{n}& =\hfill & {x}^{m+n}\hfill & & & \hfill \frac{{x}^{m}}{{x}^{n}}& =\hfill & {x}^{m-n}\hfill & & & \hfill {({x}^{m})}^{n}& =\hfill & {x}^{mn}\hfill \\ \hfill {x}^{-n}& =\hfill & \frac{1}{{x}^{n}}\hfill & & & \hfill {(xy)}^{n}& =\hfill & {x}^{n}{y}^{n}\hfill & & & \hfill {(\frac{x}{y})}^{n}& =\hfill & \frac{{x}^{n}}{{y}^{n}}\hfill \\ \hfill {x}^{1\text{/}n}& =\hfill & \sqrt[n]{x}\hfill & & & \hfill \sqrt[n]{xy}& =\hfill & \sqrt[n]{x}\sqrt[n]{y}\hfill & & & \hfill \sqrt[n]{\frac{x}{y}}& =\hfill & \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\hfill \\ \hfill {x}^{m\text{/}n}& =\hfill & \sqrt[n]{{x}^{m}}={(\sqrt[n]{x})}^{m}\hfill & & & & & & & & & & \end{array}[/latex]

Special Factorizations

[latex]\begin{array}{ccc}\hfill {x}^{2}-{y}^{2}& =\hfill & (x+y)(x-y)\hfill \\ \hfill {x}^{3}+{y}^{3}& =\hfill & (x+y)({x}^{2}-xy+{y}^{2})\hfill \\ \hfill {x}^{3}-{y}^{3}& =\hfill & (x-y)({x}^{2}+xy+{y}^{2})\hfill \end{array}[/latex]

Quadratic Formula

If [latex]a{x}^{2}+bx+c=0,[/latex] then [latex]x=\frac{-b±\sqrt{{b}^{2}-4ca}}{2a}.[/latex]

Binomial Theorem

[latex]{(a+b)}^{n}={a}^{n}+(\begin{array}{l}n\\ 1\end{array}){a}^{n-1}b+(\begin{array}{l}n\\ 2\end{array}){a}^{n-2}{b}^{2}+\cdots +(\begin{array}{c}n\\ n-1\end{array})a{b}^{n-1}+{b}^{n},[/latex]

where [latex](\begin{array}{l}n\\ k\end{array})=\frac{n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 3\cdot 2\cdot 1}=\frac{n!}{k!(n-k)!}[/latex]