1. **State the problem:** Simplify the expression $$\frac{((-4)^6)^2 \times (-5^2)^3}{(-4)^6 \times (-5)^2}$$. 2. **Recall the exponent rules:** - Power of a power: $$(a^m)^n = a^{m \times n}$$ - Product of powers with the same base: $$a^m \times a^n = a^{m+n}$$ - Quotient of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Apply power of a power:** $$((-4)^6)^2 = (-4)^{6 \times 2} = (-4)^{12}$$ $$(-5^2)^3 = (-5^2)^3 = (-5)^{2 \times 3} = (-5)^6$$ 4. **Rewrite the expression:** $$\frac{(-4)^{12} \times (-5)^6}{(-4)^6 \times (-5)^2}$$ 5. **Use quotient of powers:** $$(-4)^{12} \div (-4)^6 = (-4)^{12-6} = (-4)^6$$ $$(-5)^6 \div (-5)^2 = (-5)^{6-2} = (-5)^4$$ 6. **Simplify the expression:** $$(-4)^6 \times (-5)^4$$ 7. **Calculate powers:** $$(-4)^6 = (4)^6 = 4096^2 = 4096 \times 4096 = 16777216$$ (since even power makes negative base positive) $$(-5)^4 = (5)^4 = 625$$ 8. **Multiply the results:** $$16777216 \times 625 = 10485760000$$ **Final answer:** $$10485760000$$