Power
ArithmeticA power is the result of multiplying a base by itself a specified number of times, written as base^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 base^exponent is called a power.
Example
$3^4 = 81$ is a power. Read it as "$3$ to the fourth power." $10^2 = 100$ is "$10$ to the second power," or "$10$ squared."
Key Insight
Powers of $10$ are especially important: $10^1=10$, $10^2=100$, $10^3=1{,}000$. Each power of $10$ adds a zero, which is the basis of our whole number system.
Definition
A power $b^n$ is a shorthand for repeated multiplication. Special names: $n=2$ is "squared," $n=3$ is "cubed," $n=1$ is the number itself, $n=0$ is $1$ (for $b\neq0$). Powers of $2$: $1, 2, 4, 8, 16, 32, \ldots$ Powers of $10$ define place values.
Example
Powers of $2$ are used in computing: $2^8 = 256$ distinct values in a byte. Powers of $10$ define metric prefixes: $10^3 =$ kilo, $10^6 =$ mega, $10^9 =$ giga.
Key Insight
Doubling problems always involve powers of $2$. The story of the chessboard and grains of rice illustrates how fast powers grow: $2^{64} - 1 \sim 1.8 \times 10^{19}$ grains, far exceeding any real supply.
Definition
In algebra, a power monomial $x^n$ is a basis element of the polynomial ring $R[x]$. Power functions $f(x) = x^n$ have derivatives $f'(x) = n \cdot x^{n-1}$ (power rule). Complex powers $z^w = e^{w \operatorname{Log} z}$ are multivalued. In group theory, $g^n$ denotes repeated application of group operation $n$ times.
Example
Power series: $\sum_{n=0}^{\infty} a_n x^n$. The function $e^x$ has the power series $1 + x + x^2/2! + x^3/3! + \ldots$, converging for all real (and complex) $x$.
Key Insight
Power series are the universal language of analytic functions. Any smooth function expressible as a power series can be differentiated and integrated term-by-term, making power series central to differential equations, physics, and engineering.