# If x^{2} =y^{3} and y>1, what does x^frac{2}{3} equal in terms of y?

Question
Equations and inequalities
If $$\displaystyle{x}^{{{2}}}={y}^{{{3}}}$$ and y>1, what does $$\displaystyle{x}^{{\frac{{{2}}}{{{3}}}}}$$ equal in terms of y?

2021-01-20

$$(X)^{2}=(Y)^{3}\ \&\ Y>1(given)$$
$$\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={\left({X}^{{{2}}}\right)}^{{\frac{{{1}}}{{{3}}}}}-------------{r}\underline{{e}}$$
$$(X)^{\frac{m}{n}}=(X)^{m\frac{1}{n}}$$
compensate by $$( X )^{2} = ( Y )^{3}$$
$$\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={\left({Y}^{{{3}}}\right)}^{{\frac{{{1}}}{{{3}}}}}={\left({Y}\right)}^{{3}}\times{\frac{{{1}}}{{{3}}}}={Y}$$
thus
$$\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}$$
$$\displaystyle{\left({X}\right)}^{{{2}}}={\left({Y}\right)}^{{{3}}}$$ ------------ take the cube root of both sides
$$\displaystyle{\left({X}^{{{2}}}\right)}^{{\frac{{{1}}}{{{3}}}}}={\left({\left({Y}\right)}^{{{3}}}\right)}^{{\frac{{{1}}}{{{3}}}}}$$
$$\displaystyle{\left({X}^{{{2}\times{\frac{{{1}}}{{{3}}}}}}={Y}^{{{3}}}\times{\frac{{{1}}}{{{3}}}}\right.}$$
$$\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}$$
answer ------------- $$\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}$$

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