Exponentiation and Power¶

Definition¶

In the expression $x^\alpha\text{,}$ the number $x$ is the base, $\alpha$ is the exponent, and $x^\alpha$ is a power of $x\text{.}$ The meaning is introduced in stages:

For a positive integer $n\text{,}$ $x^n$ is the product of $n$ factors equal to $x\text{.}$ We use $x^0=1\text{;}$ in the algebraic convention of this course, this includes $0^0=1\text{.}$
If $x\neq0$ and $n$ is a positive integer, then $x^{-n}=1/x^n\text{.}$
If $x>0$ and $q=a/b$ is rational with $a\in\mathbb{Z}$ and $b\geq1\text{,}$ then $x^q=(x^{1/b})^a\text{.}$
If $x>0$ and $\alpha$ is any real number, then $x^\alpha$ is defined and obeys the usual exponent laws.

For a negative base, arbitrary real powers are not generally real. When $q=p/r$ is in lowest terms with odd denominator $r\text{,}$ the course uses $x^{p/r}=(\sqrt[r]{x})^p$ for every real $x$ (and for $x\neq0$ when $p<0$ $0$ ).

Interpretation¶

For positive integer exponents, exponentiation describes repeated multiplication: $x^n$ multiplies $n$ factors equal to $x$ . Each later stage of the Definition extends this meaning in the only way that keeps the exponent laws true.

The laws determine the extensions. From $x^nx^0=x^n$ we need $x^0=1$ $1$ , and from $x^nx^{-n}=x^0=1$ we need $x^{-n}=1/x^n$ $1/ x^{n}$ , the reciprocal of $x^n$ . From $(x^{1/b})^b=x$ we need $x^{1/b}$ to be the $b$ -th root of $x$ . An irrational exponent is then approached through rational ones: $2^{\sqrt{2}}$ is the limit of $2^{1.4},2^{1.41},2^{1.414},\ldots$ .

The domain of a power matters: even roots require nonnegative inputs, odd roots allow negative inputs, and an irrational exponent requires a positive base. This is why the Key Results are stated for $x,y>0$ $0$ . For a negative base the laws can fail: with $x=-8$ $- 8$ , $(x^2)^{1/6}=64^{1/6}=2$ $2$ , whereas $x^{2\cdot(1/6)}=x^{1/3}=-2$ $- 2$ . For the same reason the Definition writes the exponent in lowest terms before it applies a root to a negative base.

Fixing the base and varying the exponent gives the exponential function $b^x$ ; fixing the exponent and varying the base gives the power function $x^k$ . The logarithm $\log_b x$ answers the reverse question: which exponent turns $b$ into $x$ .

A common slip is to read a negative exponent as a negative number: $2^{-3}=1/8$ $1/8$ , which is positive.

Key Results¶

For positive real numbers $x,y$ and real exponents $\alpha,\beta\text{,}$ we have:

Product of powers: $x^\alpha x^\beta=x^{\alpha+\beta}\text{.}$
Power of a power: $(x^\alpha)^\beta=x^{\alpha\beta}\text{.}$
Power of a product: $(xy)^\alpha=x^\alpha y^\alpha\text{.}$
Power of a quotient: $(x/y)^\alpha=x^\alpha/y^\alpha\text{.}$
If $x>1\text{,}$ then $x^\alpha>x^\beta$ exactly when $\alpha>\beta\text{.}$
If $0<x<1\text{,}$ then $x^\alpha>x^\beta$ exactly when $\alpha<\beta\text{.}$

Examples¶

$2^3=8\text{.}$
$5^0=1\text{.}$
$4^{-3/2}=1/8\text{.}$
$(-8)^{2/3}=(\sqrt[3]{-8})^2=4\text{.}$

Prerequisites¶

Multiplication and Product

Sources¶

Math I: Calculus · Lectures 1–4

Symbols: alpha, beta, element of, integers, square root

Last updated: 20 October 2026 03:02