1. **State the problem:** Simplify the expression $$\frac{x^{7} \cdot \left(y^{-6} \cdot z^{5}\right)^{2}}{x^{8} \cdot y^{-3} \cdot z^{4}}$$ and write the answer with positive exponents only. 2. **Apply exponent rules:** Recall that $(a^m)^n = a^{m \cdot n}$ and $\frac{a^m}{a^n} = a^{m-n}$. 3. **Simplify the numerator:** $$\left(y^{-6} \cdot z^{5}\right)^2 = y^{-6 \cdot 2} \cdot z^{5 \cdot 2} = y^{-12} \cdot z^{10}$$ So numerator becomes: $$x^{7} \cdot y^{-12} \cdot z^{10}$$ 4. **Write the full fraction:** $$\frac{x^{7} \cdot y^{-12} \cdot z^{10}}{x^{8} \cdot y^{-3} \cdot z^{4}}$$ 5. **Divide like bases by subtracting exponents:** $$x^{7-8} \cdot y^{-12 - (-3)} \cdot z^{10 - 4} = x^{-1} \cdot y^{-12 + 3} \cdot z^{6} = x^{-1} \cdot y^{-9} \cdot z^{6}$$ 6. **Rewrite with positive exponents:** $$\frac{z^{6}}{x^{1} \cdot y^{9}} = \frac{z^{6}}{x \cdot y^{9}}$$ **Final answer:** $$\boxed{\frac{z^{6}}{x \cdot y^{9}}}$$