1. **State the problem:** Simplify the expression $$\frac{5^{723}}{5^{721}} \cdot \frac{3^{300}}{3^{249}} \cdot \left(\frac{3 \cdot 4^3}{3^{-2} \cdot 4^{-7}}\right)^0 \cdot \left(\frac{4 \times 10^3}{10^{-2}}\right)^2$$. 2. **Use the laws of exponents:** - $$\frac{a^m}{a^n} = a^{m-n}$$ - $$a^0 = 1$$ for any nonzero $$a$$ - $$(a^m)^n = a^{m \cdot n}$$ - $$a^m \cdot a^n = a^{m+n}$$ 3. **Simplify each part:** - $$\frac{5^{723}}{5^{721}} = 5^{723-721} = 5^2$$ - $$\frac{3^{300}}{3^{249}} = 3^{300-249} = 3^{51}$$ - $$\left(\frac{3 \cdot 4^3}{3^{-2} \cdot 4^{-7}}\right)^0 = 1$$ because anything to the zero power is 1. - Simplify inside the last parentheses: $$\frac{4 \times 10^3}{10^{-2}} = 4 \times 10^{3 - (-2)} = 4 \times 10^{5}$$ Then square it: $$\left(4 \times 10^{5}\right)^2 = 4^2 \times (10^{5})^2 = 16 \times 10^{10}$$ 4. **Combine all parts:** $$5^2 \times 3^{51} \times 1 \times 16 \times 10^{10} = 25 \times 3^{51} \times 16 \times 10^{10}$$ 5. **Multiply constants:** $$25 \times 16 = 400$$ 6. **Final simplified expression:** $$400 \times 3^{51} \times 10^{10}$$ This is the simplified form of the original expression.