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

\(\begin{array}{flushleft}
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} & \text{ Quadratic formula}\\
y = mx + b & \text{ Slope-intercept form of a straight line}\\
y - y_1 = m(x - x_1) & \text{ Point-slope form of a straight line}\\
m = \dfrac{y_2 - y_1}{x_2 - x_1} & \text{ Slope of a straight line passing through the points } (x_1, x_2) \text{ and } (y_1, y_2)
\end{array}\)