Exponent
An exponent is a number that tells how many times a base is multiplied by itself.
Formula
b^n = b \times b \times \ldots \times b \ (n \text{ times})
Definition
An exponent tells you how many times to multiply a number by itself, written as a small number above and to the right of the base. For a real base $b$ and positive integer exponent $n$, $b^n = b \times b \times \ldots \times b$ ($n$ factors), with extended rules $b^0 = 1$ (for $b \neq 0$), $b^{-n} = 1/b^n$, and $b^{1/n} = $ the $n$-th root of $b$, following laws like $b^m \times b^n = b^{m+n}$ and $(b^m)^n = b^{mn}$. For real $b > 0$ and real exponent $r$, $b^r$ is defined more generally via the exponential function, $b^r = e^{r \ln b}$, extending exponentiation to all real (and complex) exponents; $e^x$ itself is the unique function satisfying $\frac{d}{dx}(e^x) = e^x$ and $e^0 = 1$, making it central to analysis.
Example
$2^4$ means $2 \times 2 \times 2 \times 2 = 16$: the base is $2$ and the exponent is $4$, read "two to the fourth power." $3^5 = 243$, $5^0 = 1$, $2^{-3} = 1/8$, and $2^3 \times 2^4 = 2^7 = 128$ shows the rule that multiplying same-base powers adds exponents. Euler's formula, $e^{ix} = \cos(x) + i\sin(x)$, gives $e^{i\pi} + 1 = 0$, connecting the five most important constants in mathematics.
Key Insight
Exponents are a shortcut for repeated multiplication, just like multiplication is a shortcut for repeated addition; $2^{10} = 1{,}024$ would take a long time to write as repeated multiplication. The rule $b^m \times b^n = b^{m+n}$ means exponents convert multiplication into addition, the key insight behind logarithms, $\log(xy) = \log(x) + \log(y)$. Exponential growth governed by $b^n$ ($b > 1$) is the basis of compound interest, population models, radioactive decay ($b < 1$), and the spread of infections: anything that "doubles at a fixed rate" is governed by an exponent.