1. Problem: Simplify $6^4 \cdot 6^4$. 2. Use the rule for multiplying powers with the same base: $a^m \cdot a^n = a^{m+n}$. 3. Calculate the exponent sum: $4 + 4 = 8$. 4. So, $6^4 \cdot 6^4 = 6^8$. 5. Problem: Simplify $(-5)^2$. 6. Square the number: $(-5)^2 = (-5) \cdot (-5) = 25$. 7. Problem: Simplify $\frac{2^9}{2^5}$. 8. Use the rule for dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$. 9. Calculate the exponent difference: $9 - 5 = 4$. 10. So, $\frac{2^9}{2^5} = 2^4$. 11. Problem: Simplify $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3^2$. 12. Rewrite as powers: $3^1 \cdot 3^1 \cdot 3^1 \cdot 3^1 \cdot 3^2$. 13. Sum exponents: $1 + 1 + 1 + 1 + 2 = 6$. 14. So, $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3^2 = 3^6$. 15. Problem: Simplify $\frac{12^5 \cdot 12^7}{12^4}$. 16. Multiply numerator powers: $12^{5+7} = 12^{12}$. 17. Divide powers: $\frac{12^{12}}{12^4} = 12^{12-4} = 12^8$. 18. Problem: Simplify $\left(\frac{7^5}{7^7}\right)^2$. 19. Simplify inside parentheses: $\frac{7^5}{7^7} = 7^{5-7} = 7^{-2}$. 20. Apply the outer exponent: $(7^{-2})^2 = 7^{-2 \cdot 2} = 7^{-4}$. 21. Problem: Simplify $\frac{4^8}{4^5}$. 22. Divide powers: $4^{8-5} = 4^3$. 23. Problem: Simplify $(-10) \cdot (-10)^4$. 24. Rewrite as powers: $(-10)^1 \cdot (-10)^4 = (-10)^{1+4} = (-10)^5$. 25. Problem: Simplify $\frac{((-3)^4)^3}{(-3)^2}$. 26. Apply power of a power: $((-3)^4)^3 = (-3)^{4 \cdot 3} = (-3)^{12}$. 27. Divide powers: $\frac{(-3)^{12}}{(-3)^2} = (-3)^{12-2} = (-3)^{10}$. 28. Problem: Solve for $x$ in $\frac{8^x}{8^5} = 8^7$. 29. Simplify left side: $8^{x-5} = 8^7$. 30. Since bases are equal, set exponents equal: $x - 5 = 7$. 31. Solve for $x$: $x = 7 + 5 = 12$. 32. Problem: Simplify $(-11)^7 \cdot (-11)^4 = \frac{(-11)^{10}}{(-11)^3}$. 33. Left side: $(-11)^{7+4} = (-11)^{11}$. 34. Right side: $\frac{(-11)^{10}}{(-11)^3} = (-11)^{10-3} = (-11)^7$. 35. So, $(-11)^{11} = (-11)^7$ is false unless $(-11)^{11-7} = 1$, which is not true. 36. Problem: Simplify $(6^9)^{10} = \frac{(6^{12})^2}{6^4}$. 37. Left side: $(6^9)^{10} = 6^{9 \cdot 10} = 6^{90}$. 38. Right side numerator: $(6^{12})^2 = 6^{12 \cdot 2} = 6^{24}$. 39. Right side: $\frac{6^{24}}{6^4} = 6^{24-4} = 6^{20}$. 40. So, $6^{90} = 6^{20}$ is false unless $90 = 20$, which is not true. Final answers: $6^4 \cdot 6^4 = 6^8$ $(-5)^2 = 25$ $\frac{2^9}{2^5} = 2^4$ $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3^2 = 3^6$ $\frac{12^5 \cdot 12^7}{12^4} = 12^8$ $\left(\frac{7^5}{7^7}\right)^2 = 7^{-4}$ $\frac{4^8}{4^5} = 4^3$ $(-10) \cdot (-10)^4 = (-10)^5$ $\frac{((-3)^4)^3}{(-3)^2} = (-3)^{10}$ $x = 12$ for $\frac{8^x}{8^5} = 8^7$ $(-11)^7 \cdot (-11)^4 \neq \frac{(-11)^{10}}{(-11)^3}$ (not equal) $(6^9)^{10} \neq \frac{(6^{12})^2}{6^4}$ (not equal)