\documentclass{article}

\begin{document}

$$3x^2 - x + 2 = 0$$
%Use the karat symbol ^ to typeset an exponent.

$$3x^{20} - x + 2 = 0$$
%Use braces when the argument is more than one character in length.

$$2^{3^x}$$
%Use braces to raise a power to a power.

$$a_0, a_1, a_2, ..., a_{100}$$
%Use an underscore to typset a subscript.

$$A = \pi r^2$$
%use \pi to typeset the number pi.

$$\alpha^3 + \beta^2 + \gamma = 0$$
%Use \name to typeset lowercase Greek letters.

$$\Sigma$$
%Use \Name to typeset upper case Greek letters.

$$\sin^2 x + \cos^2 x = 1$$
%trig functions

$$y = \arcsin x$$
%inverse trig functions

$$y = \sin^{-1} x$$
%inverse trig functions

$$y = \log_2 x$$
%logarithms

$$y = \ln x$$
%natural logarithms

\end{document}