Exponent
ArithmeticAn 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. It is written as a small number above and to the right of the base.
Example
$2^4$ means $2 \times 2 \times 2 \times 2 = 16$. The base is $2$ and the exponent is $4$. Read it as "two to the fourth power."
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.
Definition
For a real base $b$ and positive integer exponent $n$, $b^n = b \times b \times \ldots \times b$ ($n$ factors). Extended rules: $b^0 = 1$ (for $b \neq 0$), $b^{-n} = 1/b^n$, $b^{1/n} = n\text{-th root of } b$. Laws: $b^m \times b^n = b^{m+n}$, $(b^m)^n = b^{mn}$, $(ab)^n = a^n b^n$.
Example
$3^5 = 243$. $5^0 = 1$. $2^{-3} = 1/8$. $(2^3)^2 = 2^6 = 64$. $2^3 \times 2^4 = 2^7 = 128$.
Key Insight
The rule $b^m \times b^n = b^{m+n}$ means that exponents convert multiplication into addition. This is the key insight behind logarithms: $\log(xy) = \log(x) + \log(y)$.
Definition
For real $b > 0$ and real exponent $r$, $b^r$ is defined via the exponential function: $b^r = e^{r \ln b}$. This extends exponentiation to all real (and complex) exponents. The exponential function $e^x$ is the unique function satisfying $\frac{d}{dx}(e^x) = e^x$ and $e^0 = 1$, and is the central function of analysis.
Example
$2^{\sqrt{2}} = e^{\sqrt{2} \ln 2} \approx 2.665$. Euler's formula: $e^{ix} = \cos(x) + i\sin(x)$, so $e^{i\pi} + 1 = 0$, connecting the five most important constants in mathematics.
Key Insight
The 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.