$(10)^{24} \times (10)^{-21} = $x$

### Concept & Formula This question tests the fundamental product rule of exponents. When multiplying two terms with the exact same base, you simply add their exponents together. $$ a^m \times a^n = a^{m+n} $$ ### Step-by-Step Solution * **Given:** The expression to simplify is $(10)^{24} \times (10)^{-21}$. * **Calculation:** 1. Confirm that both terms share the same base, which is $10$. 2. Apply the product rule by adding the exponents: $10^{(24 + (-21))}$ 3. Simplify the exponent arithmetic: $10^{(24 - 21)} = 10^3$ 4. Calculate the final standard numerical value: $10 \times 10 \times 10 = 1000$ ### Exam Strategy & Shortcut This is a direct application of a foundational rule. The moment you see identical bases being multiplied, immediately look at the powers. Mentally execute $24 - 21 = 3$. Knowing $10^3 = 1000$ takes only a fraction of a second. No written steps are necessary. ### Common Pitfall Students rushing through the exam might mistakenly subtract the exponents incorrectly due to the negative sign, perhaps computing $24 - (-21) = 45$, or they might multiply the bases to get $100$ and then do something with the powers. Always remember: keep the base, add the powers. ### Final Answer **Therefore, the correct answer is 1000.**