Simplifying the exponent (2/6)

Question

this is confusing me.

Adrián confirmed that $x^{2/6} = |x^{1/3}|$

But why does my Casio ClassPad say $x^{2/6} = x^{1/3}$? Why does Grapher graph $y=x^{2/6}$ like this?

Are these programs wrong?

Answer 1

I feel there are many ways to understand it, and maybe different software understands it differently which results in different value - I think the third one should be correct (i.e. Casio is correct).

1. If you understand it as $$x^{2/6} = (x^2)^{1/6} = \sqrt[6]{x^2}$$ Then $x$ could take any number, and $x^{2/6} = |x|^{1/3} = |x^{1/3}|$. This coincides with first one you mentioned in your question.

2. If you understand it as $$x^{2/6} = (x^{1/6})^2 = (\sqrt[6]{x})^2$$ Then in order for $\sqrt[6]{x}$ to exist, we need $x\ge0$, thus $x^{2/6} = x^{1/3}$ and $x\ge 0$. This coincides with the one mentioned in the comment.

3. If we always calculate the final value of the exponent (i.e. $2/6=1/3=0.3333...$), and then apply to $x$ - we get $x^{2/6} = x^{1/3}$ for $x \in \mathbb R$. I prefer this way, because if we want to express other meaning, it's better to explicitly denote as, for example, $\sqrt[6]{x^2}$. Plus there are so many ways to write $1/3$, $1/3 = 2/6 = \sqrt{2}/3\sqrt{2}=i/3i$, and thus the most unambiguous way is to calculate the final value of the exponent and then apply to $x$, unless explicitly specified. This coincides with Casio.