Simplify the expression. (x^3cdot z^-(1)/(2))^(3)/(5) Write your answer without using negative exponents. Assume that all variables are positive real numbers.

To simplify the expression \((x^{3} \cdot z^{-\frac{1}{2}})^{\frac{3}{5}}\), we will follow these steps:<br /><br />1. Apply the power of a product rule: \((ab)^m = a^m \cdot b^m\).<br />2. Apply the power of a power rule: \((a^m)^n = a^{mn}\).<br /><br />Let's start with the given expression:<br />\[<br />(x^{3} \cdot z^{-\frac{1}{2}})^{\frac{3}{5}}<br />\]<br /><br />First, apply the power of a product rule:<br />\[<br />(x^{3})^{\frac{3}{5}} \cdot (z^{-\frac{1}{2}})^{\frac{3}{5}}<br />\]<br /><br />Next, apply the power of a power rule to each term:<br />\[<br />x^{3 \cdot \frac{3}{5}} \cdot z^{-\frac{1}{2} \cdot \frac{3}{5}}<br />\]<br /><br />Simplify the exponents:<br />\[<br />x^{\frac{9}{5}} \cdot z^{-\frac{3}{10}}<br />\]<br /><br />Since we need to write the answer without using negative exponents, we can rewrite \(z^{-\frac{3}{10}}\) as \(\frac{1}{z^{\frac{3}{10}}}\):<br />\[<br />x^{\frac{9}{5}} \cdot \frac{1}{z^{\frac{3}{10}}}<br />\]<br /><br />Combine the terms into a single fraction:<br />\[<br />\frac{x^{\frac{9}{5}}}{z^{\frac{3}{10}}}<br />\]<br /><br />Thus, the simplified expression is:<br />\[<br />\boxed{\frac{x^{\frac{9}{5}}}{z^{\frac{3}{10}}}}<br />\]