Power functions: why scale changes the result

Increase a cube’s edge from 2 to 4 cm. Length doubles, one face’s area grows from 4 to 16 cm², and volume from 8 to 64 cm³. One scale change gives different output factors depending on the exponent.

Use roots and rational powers and reading graphs. For geometric scaling, multiply all corresponding lengths by the same factor so the shapes remain similar.

The exponent is fixed; the base varies

A power function has form $f(x)=kx^a$ : $k$ and $a$ are constant while $x$ varies. Examples include $x$ , $x^2$ , $x^3$ , $1/x=x^{-1}$ and $\sqrt x=x^{1/2}$ on their respective domains. In $2^x$ , the exponent varies instead: it is an exponential function, not a power function. Definition, OpenStax §5.2.

For $x>0$ and a positive scale factor $s$ , $f(sx)=s^a f(x)$ . With $a=1,2,3$ , doubling gives factors 2,4,8. With $a=-1$ , it gives half; with $a=1/2$ , it gives $\sqrt2$ , about 1.414, neither half nor double.

Parity of a positive integer exponent

For a positive integer $n$ , $x^n$ is defined on the entire real line. If $n$ is even, $(-x)^n=x^n$ : the graph is symmetric across the vertical axis and outputs are nonnegative. It decreases toward 0 from the left and increases to the right; minimum 0.

If $n$ is odd, $(-x)^n=-x^n$ : the graph is symmetric about the origin and increases on the whole line. Its range is all real numbers. This is why a cube root can be defined for negative numbers while a real square root cannot.

Five power laws

y = x
y = x²
y = x³
y = √x
y = 1/x

Shared axes, abstract units. Window: −2≤x≤2, −4≤y≤4. The square root starts at x=0; the two branches of 1/x are separate. Portions outside the window are not shown.

On $0<x<1$ , a larger positive exponent gives a smaller value: at $x=1/2$ , $x=1/2$ , $x^2=1/4$ and $x^3=1/8$ . For $x>1$ , the order reverses. “A cube is always larger than a square” is false without a domain.

A negative exponent excludes zero

$x^{-n}=1/x^n$ for $x\ne0$ . The graph of $x^{-1}$ has two branches and shares the sign of $x$ ; $x^{-2}$ has two positive branches. Neither is defined at zero. A vertical line must not join the curve across the break.

The zero exponent gives $x^0=1$ for $x\ne0$ . This course leaves $0^0$ undefined. Zero to a positive exponent is zero when that power is defined.

Roots determine a fractional exponent’s domain

For $x>0$ , integer $p$ and positive integer $q$ , define $x^{p/q}=(\sqrt[q]x)^p$ . A positive $p/q$ allows zero to be included; a negative one does not. Thus $x^{1/2}$ has domain $[0,\infty)$ and $x^{-1/2}$ has $(0,\infty)$ . The square root increases and has only nonnegative outputs. The cube root $\sqrt[3]x$ is defined on the whole line and increases too. Roots and powers, OpenStax §1.3.

Branch · A domain conventionGeneral power laws are safe for positive bases; this is a separate extension.

Rational exponents with a negative base

To allow $x<0$ , first reduce $p/q$ to lowest terms with $q>0$ . If $q$ is odd, define $x^{p/q}=(\sqrt[q]x)^p$ . For instance, $(-8)^{1/3}=-2$ and $(-8)^{2/3}=4$ . If $q$ is even, there is no real value for negative $x$ . Zero remains excluded for negative exponents.

This convention needs care: $2/6=1/3$ , so $(-8)^{2/6}=-2$ , but $(({-8})^2)^{1/6}=64^{1/6}=2$ . Regrouping powers changed the answer. Do not transfer positive-base laws to negative bases without conditions or hide the domain in exponent notation.

An exponent describes output scaling only together with a domain. Check the base’s sign and zero first, then compare scales: geometric length, area and volume should not all grow alike.