1. **State the problem:** Simplify the expression $$\left(2y^{-4} \cdot 2y^2\right)^3$$. 2. **Recall the rules:** - When multiplying terms with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$. - When raising a product to a power, raise each factor to that power: $$(ab)^n = a^n b^n$$. - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$. 3. **Simplify inside the parentheses first:** $$2y^{-4} \cdot 2y^2 = (2 \cdot 2)(y^{-4} \cdot y^2) = 4y^{-4+2} = 4y^{-2}$$ 4. **Rewrite the expression:** $$\left(4y^{-2}\right)^3$$ 5. **Apply the power to each factor:** $$4^3 \cdot \left(y^{-2}\right)^3 = 64y^{-6}$$ 6. **Express with positive exponents:** $$64y^{-6} = \frac{64}{y^6}$$ This matches option C. **Summary:** Multiply coefficients, add exponents for like bases, then raise the entire product to the power by applying the exponent to each factor and multiplying exponents accordingly. Finally, rewrite negative exponents as fractions for clarity.