1. **State the problem:** Simplify the expression $$\frac{512^3 \times 2^{-9} \div 2^8}{9^2 \div 9 - 5^0}$$. 2. **Rewrite the expression:** $$\frac{512^3 \times 2^{-9} \div 2^8}{\frac{9^2}{9} - 5^0}$$ 3. **Simplify the numerator:** - Note that $512 = 2^9$, so $512^3 = (2^9)^3 = 2^{27}$. - The numerator becomes: $$2^{27} \times 2^{-9} \div 2^8$$ - Using exponent rules for multiplication and division: $$2^{27} \times 2^{-9} = 2^{27 + (-9)} = 2^{18}$$ $$2^{18} \div 2^8 = 2^{18 - 8} = 2^{10}$$ 4. **Simplify the denominator:** - Calculate $9^2 = 81$. - Then $\frac{9^2}{9} = \frac{81}{9} = 9$. - Also, $5^0 = 1$. - So denominator is: $$9 - 1 = 8$$ 5. **Combine numerator and denominator:** $$\frac{2^{10}}{8}$$ - Since $8 = 2^3$, rewrite denominator: $$\frac{2^{10}}{2^3} = 2^{10 - 3} = 2^7$$ 6. **Final answer:** $$2^7 = 128$$