Question about rational exponents From a book, the question goes like this:
This is how John simplified $\sqrt[3]{16x^5}$.
$\sqrt[3]{16x^5} = \sqrt[3]{8} \ \cdot \sqrt[3]{2x^5} = 2\sqrt[3]{2x^5}$ 
Explain the mistake in John's work and then correct it.
My answer: Isn't John correct? I mean,
\begin{align}
\sqrt[3]{16x^5} & = \sqrt[3]{(2\cdot 8)\cdot x^5} \\
 & = (2\cdot 8 \cdot x^5)^{1/3}\\ 
 & = 2^{1/3}\cdot8^{1/3}\cdot x^{5/3} \qquad \text{from $(xy)^n = x^ny^n$ }\\ 
 & = 8^{1/3}\cdot2^{1/3}\cdot x^{5/3} \qquad \text{just rearranging}\\ 
 & = \sqrt[3]{8} \cdot (2x^5)^{1/3}\\
 & = \sqrt[3]{8} \cdot \sqrt[3]{2x^5}  \qquad \text{which is the same with John's answer}\\
 & = 2\sqrt[3]{2x^5}
\end{align}
Am I missing something here? Thanks for the help!

The question here is what does the autor consider to be a simplified expression. John didn't made any mistake, but someone would say that a more simplified expression would be
$\sqrt[3]{16 x^5} = 2^{(\dfrac{4}{3})} \cdot x^{(\dfrac{5}{3})}$.
