simplify $\sqrt[3]{x \sqrt[3]{ x \sqrt[3]{x ...}} }$ -- if $x$ is negative? Was given this recreational problem: simplify $$\sqrt[3]{x \sqrt[3]{ x \sqrt[3]{x ...}} }$$
The solution isn't hard. Let $y = \sqrt[3]{x \sqrt[3]{ x \sqrt[3]{x ...}} }$, then $y^3 = xy$, $y=\sqrt{x}$
The problem didn't specify, but if $x$ is negative, would this work? 

Let $y=\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\dots}}}$
So,
$y=\sqrt[3]{xy}$
Now raise both side to the power $3$, we get:
$y^3=xy$
Now, $y\ne0$ because $x$ is negative as mentioned in the problem statement, we can divide both sides by $y$, we get:
$y^2=x$
However $y^2=x$ has no real solution since $x$ is negative. Hence the given expression does not have a closed form. [THIS IS THE REQUIRED ANSWER].

When $x$ is positive, then the equation $y^2=x$ reduces to $y=\sqrt{x}$.
