The problem asks us to simplify the expression $\frac{\sqrt[5]{x}}{\sqrt[2]{z}}$ and rewrite it using rational exponents. We are given that the variables represent positive real numbers.

1. Problem Description

The problem asks us to simplify the expression $\frac{\sqrt[5]{x}}{\sqrt[2]{z}}$ and rewrite it using rational exponents. We are given that the variables represent positive real numbers.

2. Solution Steps

We can rewrite the radicals using rational exponents. Recall that $\sqrt[n]{a} = a^{\frac{1}{n}}$.

So, $\sqrt[5]{x} = x^{\frac{1}{5}}$ and $\sqrt[2]{z} = z^{\frac{1}{2}}$.

Therefore, $\frac{\sqrt[5]{x}}{\sqrt[2]{z}} = \frac{x^{\frac{1}{5}}}{z^{\frac{1}{2}}}$.

We can rewrite the expression as $x^{\frac{1}{5}}z^{-\frac{1}{2}}$.

3. Final Answer

The final answer is $x^{\frac{1}{5}}z^{-\frac{1}{2}}$.