1. **State the problem:** Simplify the expression $$\frac{10^4 \times 5^3}{10^2 \times 5^5}$$. 2. **Recall the laws of exponents:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Multiplication and division are performed separately for each base. 3. **Apply the exponent rule to the base 10:** $$\frac{10^4}{10^2} = 10^{4-2} = 10^2$$ 4. **Apply the exponent rule to the base 5:** $$\frac{5^3}{5^5} = 5^{3-5} = 5^{-2}$$ 5. **Rewrite the expression with simplified bases:** $$10^2 \times 5^{-2}$$ 6. **Express negative exponent as reciprocal:** $$5^{-2} = \frac{1}{5^2}$$ 7. **Final simplified expression:** $$10^2 \times \frac{1}{5^2} = \frac{10^2}{5^2}$$ 8. **Calculate the powers:** $$10^2 = 100$$ $$5^2 = 25$$ 9. **Divide the numbers:** $$\frac{100}{25} = 4$$ **Final answer:** $$4$$