1. We need to find the value of the expression $$729^8 - 3^{18} + 85$$ and then convert it to base 9 to find the number of significant zeros in its representation. 2. Note that $$729 = 3^6$$, so $$729^8 = (3^6)^8 = 3^{48}$$. 3. Rewrite the expression using powers of 3: $$3^{48} - 3^{18} + 85$$. 4. To convert this large number to base 9, recall that base 9 is $$9 = 3^2$$. Numbers in base 9 are expressed in powers of $$3^2$$. 5. We want to find the base 9 representation of: $$3^{48} - 3^{18} + 85$$. 6. Note that $$3^{48} = (3^2)^{24} = 9^{24}$$, which is 1 followed by 24 zeros in base 9. 7. Similarly, $$3^{18} = (3^2)^9 = 9^9$$. 8. So the number is: $$9^{24} - 9^9 + 85$$ in base 9. 9. In base 9, - $$9^{24}$$ is 1 followed by 24 zeros. - $$9^9$$ is 1 followed by 9 zeros. 10. We subtract $$9^9$$ (a one with 9 zeros) from $$9^{24}$$ (a one with 24 zeros), so these 9 zeros at positions 9 through 17 will be affected. 11. The subtraction will create non-zero digits in places from 9 to 17, so zeros will appear elsewhere. 12. Since $$85 < 9^2 = 81$$, adding 85 will affect the last two digits in base 9 plus carry. 13. Calculate 85 in base 9: $$85 \div 9 = 9$$ remainder $$4$$ $$9 \div 9 = 1$$ remainder $$0$$ So, $$85 = (1 \times 9^2) + (0 \times 9^1) + 4 = (104)_9$$. 14. Add 104 (base 9) to the last digits. 15. The number of significant zeros means the count of zeros in the base 9 representation that occur between non-zero digits (not leading or trailing zeros). 16. Since the subtraction and addition affect the digits near the right end and around the 9th position, the count of significant zeros is the count of zeros between the 10 zeros (positions 10 to 17). 17. The expression simplifies to there being **9 significant zeros** in the base 9 representation. Final answer: 9