Question

$\frac{ \sqrt{x \sqrt[3]{ {x}^{2} } } \: \times \sqrt[3]{ {x}^{2} } } { {( \sqrt{ {x}^{1} } )}^{3} }$

Answer 1

$\displaystyle \frac{\sqrt{x \cdot \sqrt[3]{x^2}} \cdot \sqrt[3]{x^2}}{(\sqrt x)^3}= \frac{x^{\frac{1}{2}} \cdot (x^{\frac{2}{3}})^{\frac{1}{2}} \cdot x^{\frac{2}{3}}}{(x^{\frac{1}{2}})^3}=\frac{x^{\frac{1}{2}} \cdot x^{\frac{2}{6}} \cdot x^{\frac{2}{3}}}{x^{\frac{3}{2}}}=\\ \\ =x^{\frac{1}{2}+ \frac{1}{3}+ \frac{2}{3}- \frac{3}{2}}=x^{\frac{3}{3}-\frac{2}{2}}=x^{1-1}=x^0=1$

Answer is 1.