Can I write $(x^p)^{q} = x^{pq}$, where either of $p$ or $q$ are rational.

Question

I am confused with a very basic algebra question about the following law of exponents.

We know that $(x^n)^{m} = x^{nm}$, holds true for real $x$ and integer exponents $n, m$. I want to know whether this result holds for rational exponents aswell?

For example can I write $(x^p)^{q} = x^{pq}$, where either of $p$ or $q$ are rational.

Kindly pardon me for asking a very silly question.

Thanks for the help.

Answer 1

Let us first recall the conventional definition of rational exponentiation:$$x^{\frac{m}{n}} = \left ( x^{\frac{1}{n}} \right )^m = \left ( \sqrt[n]{x} \right ) ^m,$$where $m, n \in \mathbb{Z}$ and $x \in \mathbb{R}$ ($x \neq 0$ if need be). So, $x^{\frac{m}{n}}$ is defined whenever $\sqrt[n]{x}$ is defined ($x\neq 0$ if need be).

It should be noted that, according to the definition, we do not have the same result for different representations of a rational exponent; that is, $\frac{m}{n} = \frac{m'}{n'}$ does not necessarily imply that$$x^{\frac{m}{n}} = x^{\frac{m'}{n'}}$$($x\neq 0$ if need be) because one of the sides may not be defined, for example, while $\frac{1}{3} = \frac{2}{6}$, we do not have the identity$$(-1)^{\frac{1}{3}} = (-1)^{\frac{2}{6}},$$noting that the right hand side is not defined since $\sqrt[6]{-1}$ is not defined.

Now, suppose that $x^p = x^{\frac{m}{n}}$ and $x^q = x^{\frac{r}{s}}$, where $m, n, r, s \in \mathbb{Z}$ and $x\in \mathbb{R}$ ($x\neq 0$ if need be), are defined. So we have \begin{align*}\left (x^p \right )^q & = \left ( x^{\frac{m}{n}} \right )^{\frac{r}{s}}\\ & = \left ( \left ( \sqrt[n]{x} \right )^m \right )^{\frac{r}{s}}\\ & = \left (\sqrt[s]{\left ( \sqrt[n]{x} \right )^m} \right )^r\\ & = \left ( \left ( \sqrt[s]{\sqrt[n]{x}} \right )^m \right )^r\\ & = \left (\sqrt[ns]{x} \right )^{mr}\\ & = x^{\frac{m}{n} \frac{r}{s}}\\ & = x^{pq} \end{align*} In the derivation above we used the following facts:

• the fact that$$\left (x^m \right )^n = x^{mn},$$where $m, n \in \mathbb{Z}$ and both sides are defined, which can be easily shown by induction;
• the fact that$$\sqrt[n]{\sqrt[m]{x}}=\sqrt[mn]{x},$$where $m, n \in \mathbb{Z}$ and both sides are defined, which follows directly from the previous fact and the definition of the $n$th root of a real number;
• the fact that$$\sqrt[n]{x^m}=\left (\sqrt[n]{x} \right )^m,$$where $m, n \in \mathbb{Z}$ and both sides are defined, which follows directly from the definition of the $n$th root of a real number.

Please note that although it is correct to say that the identity holds for positive real numbers $x$, from the derivation above we see that the identity holds whenever all the terms involved in the derivation, namely, $\sqrt[n]{x}$ and $\sqrt[s]{\sqrt[n]{x}}=\sqrt[ns]{x}$, are defined; for example, the identity$$\left ( (-1)^{\frac{1}{3}} \right )^{\frac{3}{5}} = (-1)^{\frac{1}{5}}$$holds because the terms $\sqrt[3]{-1}$ and $\sqrt[5]{\sqrt[3]{-1}}$ are defined.

P.S. No question at any level which is asked to gain knowledge is silly.

Answer 2

As long as x is positive, the rule holds true for rational exponents p and q. When x is negative, you run into issues with complex numbers.