1. **State the problem:** Simplify the expression $$\frac{(3^4\cdot 5^2-7^3)+(12^2-8^2)}{2^3}+\sqrt{144}-\frac{6!}{5!}$$. 2. **Recall formulas and rules:** - Exponentiation: $a^b$ means $a$ multiplied by itself $b$ times. - Factorial: $n! = n \times (n-1) \times \cdots \times 1$. - Square root: $\sqrt{x}$ is the number which squared gives $x$. - Order of operations: parentheses, exponents, multiplication/division, addition/subtraction. 3. **Calculate powers and factorials:** - $3^4 = 81$ - $5^2 = 25$ - $7^3 = 343$ - $12^2 = 144$ - $8^2 = 64$ - $2^3 = 8$ - $6! = 720$ - $5! = 120$ 4. **Substitute values:** $$\frac{(81 \cdot 25 - 343) + (144 - 64)}{8} + \sqrt{144} - \frac{720}{120}$$ 5. **Simplify inside parentheses:** - $81 \cdot 25 = 2025$ - $2025 - 343 = 1682$ - $144 - 64 = 80$ 6. **Sum numerator:** $$1682 + 80 = 1762$$ 7. **Divide numerator by denominator:** $$\frac{1762}{8}$$ Show cancellation: $$\frac{\cancel{1762}}{\cancel{8}}$$ Since 1762 and 8 share no common factor, perform division: $$1762 \div 8 = 220.25$$ 8. **Calculate square root:** $$\sqrt{144} = 12$$ 9. **Calculate factorial division:** $$\frac{720}{120} = 6$$ 10. **Combine all parts:** $$220.25 + 12 - 6 = 226.25$$ **Final answer:** $$\boxed{226.25}$$