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

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

Question
Equations and inequalities
asked 2021-01-19
If \(\displaystyle{x}^{{{2}}}={y}^{{{3}}}\) and y>1, what does \(\displaystyle{x}^{{\frac{{{2}}}{{{3}}}}}\) equal in terms of y?

Answers (1)

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
\(\displaystyle{\left({X}\right)}^{{\frac{{{2}}}{{{3}}}}}={Y}\)
another answer
\(\displaystyle{\left({X}\right)}^{{{2}}}={\left({Y}\right)}^{{{3}}}\) ------------ 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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