1. **State the problem:** Simplify the expression $$\left(\frac{z^6}{4ab^{-2}}\right) \left(3a^2 b^5\right)^2$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^m / a^n = a^{m-n}$$. - When raising a power to another power, multiply exponents: $$(a^m)^n = a^{mn}$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - When multiplying terms, multiply coefficients and variables separately. 3. **Simplify the denominator:** $$4ab^{-2} = 4a \times b^{-2} = \frac{4a}{b^2}$$. 4. **Rewrite the first fraction:** $$\frac{z^6}{4ab^{-2}} = z^6 \times \frac{1}{4a} \times b^2 = \frac{z^6 b^2}{4a}$$. 5. **Simplify the second factor:** $$(3a^2 b^5)^2 = 3^2 \times (a^2)^2 \times (b^5)^2 = 9a^{4}b^{10}$$. 6. **Multiply the two factors:** $$\frac{z^6 b^2}{4a} \times 9a^{4}b^{10} = \frac{9 z^6 b^2 a^{4} b^{10}}{4a}$$. 7. **Combine like terms:** $$= \frac{9 z^6 a^{4} b^{12}}{4a}$$. 8. **Simplify the fraction by canceling $a$:** $$= \frac{9 z^6 \cancel{a^{4}} b^{12}}{4 \cancel{a}} = \frac{9 z^6 a^{3} b^{12}}{4}$$. **Final answer:** $$\boxed{\frac{9 z^6 a^{3} b^{12}}{4}}$$