Power functions with positive fractional exponents

Basic functions: Power functions

Power functions with positive fractional exponents

The best known power function with a fractional exponent is the square root $x\mapsto \sqrt{x}$ ; this can also be written in power notation because

$\sqrt{x}=x^{\frac{1}{2}}$ For a higher power root we can also convert the root notation into power notation so that it is clear that it is actually a power function:

$\sqrt[n]{x}=x^{\frac{1}{n}}\quad\text{for }n=3,4,5,\ldots$ Through the calculation rules we can also define power functions that have a positive fraction as exponent: for an irreducible fraction $\tfrac{t}{n}$ we agree on the following:

$x^{\frac{t}{n}}=\left(x^{\frac{1}{n}}\right)^t=\left(\sqrt[n]{x}\right)^t$ One can of course also choose

$x^{\frac{t}{n}}=\left(x^{t}\right)^{\frac{1}{n}}=\sqrt[n]{x^{t}}$ because it does not matter for the result.

Why the requirement of irreducibility of a fraction as an exponent You may wonder why we emphasize an irreducible fraction as an exponent. Well, if we don't do this, we'll get weird situations. The following example illustrates this.

For the power function $x\mapsto x^\frac{1}{2}$ , the domain consists of nonnegative numbers; there is no real number that can equal $(-2)^{\frac{1}{2}}=\sqrt{-2}$ . But the power function $x\mapsto x^\frac{2}{4}$ would exist for all real numbers with the above definition. For example, $(-2)^{\frac{2}{4}}=\left((-2)^{2}\right)^{\frac{1}{4}}=4^{\frac{1}{4}}=\sqrt[4]{4}$ .
Because $\tfrac{2}{4}=\tfrac{1}{2}$ we would then have two different results for equivalent fractions and we cannot accept that in mathematics. To keep the mathematical structure consistent, we have to agree that exponents with fractions are first fuly simplified.

Some examples of power function with a fraction as exponent and some rewritings in square root notation:

$\begin{aligned}x^{\frac{2}{3}}&=\sqrt[3]{x^2}\\[0.25cm] x^{\frac{3}{2}}&=\sqrt{x^3}\\ &=x\sqrt{x}\\[0.25cm] x^{\frac{5}{3}}&=\sqrt[3]{x^5}\\ &=x\sqrt[3]{x^2}\\ &=\frac{x^2}{\sqrt[3]{x}}\end{aligned}$

Below are the graphs of various power functions with fractions as exponents. The function rules are listed with the graphs.