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To simplify the expression \( \frac{1}{z^{8}} \cdot \frac{1}{z^{7}} \), we can combine the fractions:
\[
\frac{1}{z^{8}} \cdot \frac{1}{z^{7}} = \frac{1 \cdot 1}{z^{8} \cdot z^{7}} = \frac{1}{z^{8 + 7}} = \frac{1}{z^{15}}.
\]
Next, to express this in exponential form with positive exponents only, we can rewrite \( \frac{1}{z^{15}} \) as:
\[
z^{-15}.
\]
If we need to express this with positive exponents according to the prompt's instructions, we leave it in the form \(\frac{1}{z^{15}}\).
However, since the answer must be presented with positive exponents, we don't need to change it; thus, the answer is:
\[
\frac{1}{z^{15}}.
\]