$How to properly write rational exponents when expressed as roots?$

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$How to properly write rational exponents when expressed as roots?$

I'm aware that we can express $x^{2/3}$ as $(x^2)^{1/3} = (x^{1/3})^2$. Expressing these exponents in terms of roots is something that confuses me. Should the above expression be written as:
$$(\sqrt[3]{x})^2 \quad \text{or} \quad \sqrt[3]{x^2}$$ and what difference does it make if we write it in one way or the other?

Note: I'd also appreciate it if you could answer this question more generally and also refer to the domains of functions you mention (e.g. $\sqrt{x}$, so $x \geq 0$) in your answer.

Algebra Real Analysis

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First of all for rational powers, note that the following identity is not true always: $$(x^m)^n=x^{mn}\;\;\text{not always true}$$ Because with $m=2,n=\frac12$ The above identity clearly fails! That is $$(x^2)^{\frac12}\neq x^{2\times\frac12}$$ Instead it is equal to $|x|$ in above case. So it means converse of the equality may not be true always. So you cannot write it in the forms you expressed unless and until domain of x or nature of x is given. It depends upon question as well what you have generalised as x, what variable you are considering as x. To answer that question, more information is required. But if this is the only specific case you are looking for then $$x^{2/3}=(x^2)^{\frac13}=(x^{\frac13})^2\;\; \text{for all} \; x\in R$$

Aman R

$How to properly write rational exponents when expressed as roots?$

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