Basic Rules for Exponents

You will often work with exponents and will have to apply operations to them. You will need to know and use the following identities:

$$ \begin{array}{l l l} \textbf{Operation} & \textbf{Formula} & \textbf{Example} \\ \text{Multiplication: add exponents} & a^m \times a^n = a^{m+n} & x^2 \times x^3 = x^5 \\ \text{Dividing: subtract exponents} & \frac{a^m}{a^n} = a^{m-n} & \frac{x^8}{x^3} = x^5\\ \text{Exponentiation of exponents: multiply exponents} & (a^m)^n = a^{mn} & (x^3)^4 = x^{12} \\ \text{Power of a product: distribute power} & (ab)^m = a^nb^n & (2x)^4 = 16x \\ \text{Power of a quotient: distribute power} & (\frac{a}{b})^m = \frac{a^n}{b^n} & (\frac{x}{5})^2 = \frac{x^5}{25} \\ \text{Negative exponents: make positive, shift across quotient line} & a^{-n} = \frac{1}{a^n}& 3x^{-4} = \frac{3}{x^4} \\ \text{Exponentiation to zero: always equal to 1} & \frac{a^m}{a^m} = a^0 = 1 & \frac{x^{0}}{4} = \frac{1}{4}$\\ \end{array} $$