$ \sqrt[5]{x^3} = (\sqrt[5]{x})^3 $?

Question

$ \sqrt[5]{x^3} = (\sqrt[5]{x})^3 $ ?

I would suppose so given that $ x^{3/5} = x^{3(1/5)} = \sqrt[5]{x^3} $ or $ x^{3/5} = x^{(1/5)3} = (\sqrt[5]{x})^3 $

Answer 1

The rule that $\sqrt[n]{a^m} = (\sqrt[n]{a})^m$ is true when $a \ge 0$ and $n$ is even, or for any $a$ when $n$ is odd.

You need to be careful when dealing with even-index radicals and negative radicands. For example: $$\sqrt[2]{-1}^2 = -1$$ But $$\sqrt[2]{(-1)^2} = 1$$