Power
A power is the result of multiplying a base by itself a specified number of times, written as $\text{base}^{\text{exponent}}$.
Formula
b^n \ (\text{read: "}b\text{ to the }n\text{th power"})
Definition
A power is the answer you get when you multiply a number (the base) by itself a certain number of times; the expression $\text{base}^{\text{exponent}}$ is itself called a power, a shorthand for repeated multiplication. Special names apply for small exponents: $n=2$ is "squared," $n=3$ is "cubed," $n=1$ is the number itself, and $n=0$ is $1$ (for $b\neq0$). In algebra, a power monomial $x^n$ is a basis element of the polynomial ring $R[x]$, with power functions $f(x)=x^n$ satisfying the power rule $f'(x) = n \cdot x^{n-1}$; power series, $\sum_{n=0}^{\infty} a_n x^n$, extend this to represent functions like $e^x = 1 + x + x^2/2! + x^3/3! + \ldots$, converging for all real (and complex) $x$.
Example
$3^4 = 81$ is a power, read "$3$ to the fourth power," and $10^2 = 100$ is "$10$ squared." Powers of $2$ show up throughout computing, like $2^8 = 256$ distinct values in a byte, while powers of $10$ define metric prefixes ($10^3 = $ kilo, $10^6 = $ mega, $10^9 = $ giga) and place values in our number system. The chessboard-and-grains-of-rice story illustrates how fast powers grow: $2^{64} - 1 \sim 1.8 \times 10^{19}$ grains, far exceeding any real supply.
Key Insight
Powers of $10$ are especially important: each one adds a zero, the basis of our whole number system. Doubling problems always involve powers of $2$, growing far faster than intuition suggests. Power series are the universal language of analytic functions: any smooth function expressible as one can be differentiated and integrated term-by-term, making power series central to differential equations, physics, and engineering.