Simplifying ${2\sqrt[3]{e} \times 4\sqrt[3]{e^2}}$

Question

Simplify this expression: ${2\sqrt[3]{e} \times 4\sqrt[3]{e^2}}$

The answer is apparently ${8e}$

I can see that ${\sqrt e^2}$ is ${e}$ but I don't understand the reasoning of why the cube root is cancelled out.

Answer 1

$$\large \begin{align*} 2\sqrt[3]{e}\times 4\sqrt[3]{e^2}&= 2\times e^{1/3}\times 4\times (e^2)^{1/3}\\\\ &=2\times e^{1/3}\times 4\times e^{2/3} &\textsf{(since }(a^b)^c=a^{bc}\textsf{)}\\\\ &=2\times 4\times e^{1/3}\times e^{2/3}\\\\ &=8\times e^{(1/3)+(2/3)}&\textsf{(since }a^b\times a^c=a^{b+c}\textsf{)}\\\\ &=8\times e^1\\\\ &=8e \end{align*}$$

Answer 2

Hint :

$\sqrt[n]{e}=e^{1/n}$

$a^b \cdot a^c=a^{b+c}$