In order to simplify this expression, we can use the following properties:
[tex]\begin{gathered} \frac{a^b}{a^c}=a^{b-c} \\ (a^b)^c=a^{b\cdot c} \end{gathered}[/tex]
So we have:
[tex]\begin{gathered} (\frac{2x^{3/4}y^{-6}z^7}{3x^4y^{-5/2}z^{-1/2}})^{-4} \\ =\frac{x^{(3/4)\cdot(-4)}y^{(-6)\cdot(-4)}z^{7\cdot(-4)}}{3^{-4}x^{4\cdot(-4)}y^{(-5/2)\cdot(-4)}z^{(-1/2)(-4)_{}}} \\ =\frac{x^{-3}y^{24}z^{-28}}{3^{-4}x^{-16}y^{10}z^2} \\ =3^4x^{-3-(-16)}y^{24-10}z^{-28-2} \\ =81x^{13}y^{14}z^{-30} \\ =\frac{81x^{13}y^{14}}{z^{30}} \end{gathered}[/tex]
