Simplify the expression. \[ \left(z^{-\frac{5}{3}} \cdot x^{2}\right)^{\frac{1}{2}} \] Write your answer without using negative exponents. Assume that all variables are positive real numbers.

To simplify the expression \(\left(z^{-\frac{5}{3}} \cdot x^{2}\right)^{\frac{1}{2}}\), we can start by applying the exponent to each component inside the parentheses: \[ = \left(z^{-\frac{5}{3}}\right)^{\frac{1}{2}} \cdot \left(x^{2}\right)^{\frac{1}{2}} \] Now, we simplify each part individually: 1. For \(\left(z^{-\frac{5}{3}}\right)^{\frac{1}{2}}\): \[ = z^{-\frac{5}{3} \cdot \frac{1}{2}} = z^{-\frac{5}{6}} \] 2. For \(\left(x^{2}\right)^{\frac{1}{2}}\): \[ = x^{2 \cdot \frac{1}{2}} = x^{1} = x \] Now we combine these results: \[ = z^{-\frac{5}{6}} \cdot x \] Since we want to express our answer without negative exponents, we rewrite \(z^{-\frac{5}{6}}\) as \(\frac{1}{z^{\frac{5}{6}}}\): \[ = \frac{x}{z^{\frac{5}{6}}} \] Thus, the simplified expression is: \[ \frac{x}{z^{\frac{5}{6}}} \]