Answer
$y\sqrt[6]{y}$
Work Step by Step
Consider expression $\sqrt{y}\sqrt[3]{{{y}^{2}}}$,
Convert to the exponential notation,
$\sqrt{y}\sqrt[3]{{{y}^{2}}}={{y}^{\frac{1}{2}}}\times {{y}^{\frac{2}{3}}}$
Add exponent,
$\begin{align}
& {{y}^{\frac{1}{2}+\frac{2}{3}}}={{y}^{\frac{3+4}{6}}} \\
& ={{y}^{^{\frac{7}{6}}}}
\end{align}$
Convert back to radical notation,
${{y}^{^{\frac{7}{6}}}}=\sqrt[6]{{{y}^{7}}}$
Simplify the expression,
$\sqrt[6]{{{y}^{7}}}=\sqrt[6]{{{y}^{6}}}\times \sqrt[6]{y}$
Find the sixth root. we assume $x\ge 0$,
$y\sqrt[6]{y}$
Partial check:
$\begin{matrix}
{{\left( y\sqrt[6]{y} \right)}^{6}}\overset{?}{\mathop{=}}\,{{\left( y \right)}^{6}}{{\left( \sqrt[6]{y} \right)}^{6}} \\
\overset{?}{\mathop{=}}\,{{y}^{6}}\times y \\
={{y}^{7}} \\
\end{matrix}$
Thus, the expression $\sqrt{y}\sqrt[3]{{{y}^{2}}}$ can be simplified as $y\sqrt[6]{y}$
