The given expression is:
[tex](y^{\frac{3}{2}}x^{-\frac{1}{2}})^4[/tex]Note from the rule of indices that:
[tex](ab)^c=a^cb^c[/tex]Applying this rule to the given expression, we have:
[tex]\begin{gathered} y^{4(\frac{3}{2})}\times x^{-\frac{1}{2}(4)} \\ \end{gathered}[/tex]This can be further simplified as:
[tex]\begin{gathered} y^{2(3)}\times x^{-2} \\ y^6x^{-2} \end{gathered}[/tex]Therefore, the simplified expression is:
[tex]\frac{y^{6}}{x^{2}}[/tex]