1. **State the problem:** Simplify the expression $$\left(\frac{w^{-15} y^{12}}{-64 x^{3}}\right)^{-\frac{1}{3}}$$. 2. **Recall the exponent rules:** - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. - The cube root corresponds to the exponent $-\frac{1}{3}$. 3. **Apply the negative exponent:** $$\left(\frac{w^{-15} y^{12}}{-64 x^{3}}\right)^{-\frac{1}{3}} = \left(\frac{-64 x^{3}}{w^{-15} y^{12}}\right)^{\frac{1}{3}}$$ 4. **Rewrite the fraction inside the cube root:** $$= \left(-64 x^{3}\right)^{\frac{1}{3}} \div \left(w^{-15} y^{12}\right)^{\frac{1}{3}}$$ 5. **Simplify each part:** - Cube root of $-64 x^{3}$: $$\sqrt[3]{-64} = -4$$ $$\sqrt[3]{x^{3}} = x$$ So, $$\left(-64 x^{3}\right)^{\frac{1}{3}} = -4x$$ - For the denominator: $$\left(w^{-15}\right)^{\frac{1}{3}} = w^{-15 \times \frac{1}{3}} = w^{-5}$$ $$\left(y^{12}\right)^{\frac{1}{3}} = y^{12 \times \frac{1}{3}} = y^{4}$$ 6. **Combine denominator terms:** $$w^{-5} y^{4} = \frac{y^{4}}{w^{5}}$$ 7. **Put it all together:** $$\frac{-4x}{\frac{y^{4}}{w^{5}}} = -4x \times \frac{w^{5}}{y^{4}} = \frac{-4 x w^{5}}{y^{4}}$$ **Final answer:** $$\boxed{\frac{-4 x w^{5}}{y^{4}}}$$