1. **Statement of the problem:** Simplify the expression $$B = \frac{(-4)^3 \times (-6)^4 \times 2^5 \times (-27)^{-2}}{(-9)^3 \times 6^8} \times (-18)^{-4}$$ 2. **Rewrite each term using prime factorization:** - $-4 = -1 \times 2^2$ - $-6 = -1 \times 2 \times 3$ - $2$ is prime - $-27 = -1 \times 3^3$ - $-9 = -1 \times 3^2$ - $6 = 2 \times 3$ - $-18 = -1 \times 2 \times 3^2$ 3. **Express powers:** $$(-4)^3 = (-1)^3 \times 2^{6} = -1 \times 2^{6}$$ $$(-6)^4 = (-1)^4 \times 2^{4} \times 3^{4} = 1 \times 2^{4} \times 3^{4}$$ $$2^5 = 2^5$$ $$(-27)^{-2} = (-1)^{-2} \times 3^{-6} = 1 \times 3^{-6}$$ $$(-9)^3 = (-1)^3 \times 3^{6} = -1 \times 3^{6}$$ $$6^8 = 2^{8} \times 3^{8}$$ $$(-18)^{-4} = (-1)^{-4} \times 2^{-4} \times 3^{-8} = 1 \times 2^{-4} \times 3^{-8}$$ 4. **Substitute back into B:** $$B = \frac{(-1) \times 2^{6} \times 1 \times 2^{4} \times 3^{4} \times 2^{5} \times 3^{-6}}{(-1) \times 3^{6} \times 2^{8} \times 3^{8}} \times 2^{-4} \times 3^{-8}$$ 5. **Combine like terms:** Numerator powers: - $2^{6+4+5} = 2^{15}$ - $3^{4-6} = 3^{-2}$ - $(-1)^1 = -1$ Denominator powers: - $2^{8}$ - $3^{6+8} = 3^{14}$ - $(-1)^1 = -1$ 6. **Simplify the fraction:** $$B = \frac{-1 \times 2^{15} \times 3^{-2}}{-1 \times 2^{8} \times 3^{14}} \times 2^{-4} \times 3^{-8} = \frac{2^{15} \times 3^{-2}}{2^{8} \times 3^{14}} \times 2^{-4} \times 3^{-8}$$ 7. **Subtract exponents for division:** $$2^{15-8} = 2^{7}$$ $$3^{-2-14} = 3^{-16}$$ So, $$B = 2^{7} \times 3^{-16} \times 2^{-4} \times 3^{-8} = 2^{7-4} \times 3^{-16-8} = 2^{3} \times 3^{-24}$$ 8. **Final simplified form:** $$B = \frac{2^{3}}{3^{24}} = \frac{8}{3^{24}}$$ **Answer:** $$B = \frac{8}{3^{24}}$$