A coefficient that multiplies a radical expression

Question

By what rules or axioms can you get this following answer?

$k \sqrt[3]{k^2 + 1}$ Equal to $\sqrt[3]{k^5+k^3}$ ?

Answer 1

Note $k=\sqrt[3]{k^3}$. Unlike the square root, there is no ambiguity here since the cube function is a bijection. In addition, note $\sqrt[3]{x} \cdot \sqrt[3]{y}= \sqrt[3]{xy}$. This is clear by cubing $\sqrt[3]{x} \cdot \sqrt[3]{y}$ and noting cube roots are unique.

It thus follows that $k \sqrt[3]{k^2 + 1} = \sqrt[3]{k^3} \sqrt[3]{k^2 + 1} = \sqrt[3]{k^3(k^2 + 1)} = \sqrt[3]{k^5+k^3}$