To simplify the expression \( \frac{1}{x^8} \cdot \frac{1}{z^7} \), we can rewrite it as:
\[ \frac{1}{x^8 z^7} \]
This can also be expressed in exponential form with positive exponents as:
\[ x^{-8} z^{-7} \]
Thus, the fully simplified expression in exponential form with positive exponents is:
\[ x^{-8} z^{-7} \]