12.3: Important and Useful Rules/Formulas
Exponents (Assume each expression is defined.)
\(a^na^m = a^{n + m}\)
\(\dfrac{a^n}{a^m} = a^{n - m}\)
\((a^n)^m = a^{nm}\)
\((ab)^n = a^nb^n\)
\(a^{-1} = \dfrac{1}{n}\)
\(a^{-n} = \dfrac{1}{a^n}\)
\(a^0 = 1\)
\((\dfrac{a}{b})^n = \dfrac{a^n}{b^n}\)
Factorization and special product formulas
\(ab + ac = a(b + c)\)
\(a^2 + 2a +b^2 = (a+b)^2\)
\(a^2 - b^2 = (a + b)(a - b)\)
\(a^2 - 2ab + b^2 = (a-b)^2\)
Formulas
\(\begin{array}{flushleft}
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} & \text{ Quadratic formula}\\
y = mx + b & \text{ Slope-intercept form of a straight line}\\
y - y_1 = m(x - x_1) & \text{ Point-slope form of a straight line}\\
m = \dfrac{y_2 - y_1}{x_2 - x_1} & \text{ Slope of a straight line passing through the points } (x_1, x_2) \text{ and } (y_1, y_2)
\end{array}\)