The value of $112 \times 5^4$ is

### Concept & Strategy Multiplying by powers of 5 can be vastly simplified by converting the 5 into 10/2. This converts a difficult multiplication problem into a simple division problem followed by appending zeros. $$a \times 5^n = a \times \left(\frac{10}{2}\right)^n = \frac{a \times 10^n}{2^n}$$ ### Step-by-Step Solution * Recognize that $5^4$ is equal to $(10/2)^4$: $$5^4 = \frac{10^4}{2^4} = \frac{10000}{16}$$ * Substitute this fraction back into the original expression: $$112 \times \frac{10000}{16}$$ * Rearrange to perform the division first: $$\left(\frac{112}{16}\right) \times 10000$$ * Divide 112 by 16: $$112 \div 16 = 7$$ * Multiply by 10000: $$7 \times 10000 = 70000$$ ### Exam Strategy & Shortcut You should memorize the powers of 2 up to $2^{10}$ and basic multiples of 16. When you see $\times 5^4$, instantly think "divide by 16, add four zeros". $112 \div 16 = 7$. Appending four zeros gives 70000. The entire problem takes less than 5 seconds without writing anything down. ### Common Pitfall Manually calculating $5^4 = 625$ and then multiplying $112 \times 625$. While 625 is not extremely large, multiplying a three-digit number by a three-digit number manually wastes valuable time. ### Final Answer **Therefore, the correct answer is 70000.**