Exponents — The Basics.

An exponent says how many times to multiply a base by itself. $5^3 \;=\; 5 \cdot 5 \cdot 5 \;=\; 125$ The $5$ is the base, the $3$ is the exponent or power.

The power rules

These three rules cover most of what you'll do with exponents:

Product rule — same base, add exponents

$x^a \cdot x^b \;=\; x^{a+b}$ $2^3 \cdot 2^4 \;=\; 2^{3+4} \;=\; 2^7 \;=\; 128$

Quotient rule — same base, subtract exponents

$\frac{x^a}{x^b} \;=\; x^{a-b}$ $\frac{5^6}{5^2} \;=\; 5^{6-2} \;=\; 5^4 \;=\; 625$

Power of a power — multiply exponents

$(x^a)^b \;=\; x^{a \cdot b}$ $(3^2)^4 \;=\; 3^{2 \cdot 4} \;=\; 3^8$

Two special cases people forget

Anything to the zero power equals 1

$x^0 \;=\; 1 \quad (\text{for any } x \neq 0)$

Why? Because $\frac{x^3}{x^3} = x^{3-3} = x^0$, and any number divided by itself is $1$.

Negative exponent = reciprocal

$x^{-n} \;=\; \frac{1}{x^n}$ $2^{-3} \;=\; \frac{1}{2^3} \;=\; \frac{1}{8}$

A negative exponent does not make the number negative. It flips the fraction.