4)) Simplify. Assume all variables are positive. \( \left(\frac{-2 w x}{w^{-\frac{3}{4}}}\right)^{9} \) 1) Write your answer in the form A or \( \frac{A}{B^{\prime}} \) where \( A \) and B are constants or variable expressions that have no variables in common. All exponents in your answer should positive.

To simplify \( \left(\frac{-2 w x}{w^{-\frac{3}{4}}}\right)^{9} \), first, simplify the expression inside the parentheses. We know that dividing by a negative exponent is equivalent to multiplying by the positive exponent: \[ \frac{-2wx}{w^{-\frac{3}{4}}} = -2wx \cdot w^{\frac{3}{4}} = -2 w^{1 + \frac{3}{4}} x = -2 w^{\frac{7}{4}} x \] Now, raise this entire expression to the 9th power: \[ \left(-2 w^{\frac{7}{4}} x\right)^{9} = (-2)^{9} \cdot (w^{\frac{7}{4}})^{9} \cdot x^{9} \] Calculating each part: 1. \((-2)^9 = -512\) 2. \((w^{\frac{7}{4}})^{9} = w^{\frac{63}{4}}\) (multiply exponents: \(\frac{7}{4} \cdot 9 = \frac{63}{4}\)) 3. \(x^{9}\) remains as \(x^{9}\). So, we combine all parts to get: \[ -512 w^{\frac{63}{4}} x^{9} \] The final answer is: \[ \boxed{-512 w^{\frac{63}{4}} x^{9}} \] In the required form, we can represent it as: \[ \frac{-512 x^{9}}{w^{-\frac{63}{4}}} \] with no common variables in the numerator and denominator. All exponents are positive, satisfying the conditions given.