Simplify Expression Fef6B3

1. **State the problem:** Simplify the expression $$(-2k^{-3})^2(6k^2)^4(9k^4)^2$$. 2. **Recall the exponent rules:** - $(ab)^n = a^n b^n$ - $(a^m)^n = a^{mn}$ - When multiplying like bases, add exponents: $a^m \cdot a^n = a^{m+n}$ 3. **Apply the power to each term:** $$(-2k^{-3})^2 = (-2)^2 (k^{-3})^2 = 4 k^{-6}$$ $$ (6k^2)^4 = 6^4 (k^2)^4 = 1296 k^8$$ $$ (9k^4)^2 = 9^2 (k^4)^2 = 81 k^8$$ 4. **Multiply all parts together:** $$4 k^{-6} \times 1296 k^8 \times 81 k^8 = (4 \times 1296 \times 81) k^{-6 + 8 + 8}$$ 5. **Calculate the coefficients:** $$4 \times 1296 = 5184$$ $$5184 \times 81 = 419904$$ 6. **Add the exponents of $k$:** $$-6 + 8 + 8 = 10$$ 7. **Final simplified expression:** $$419904 k^{10}$$