$$ \frac{343 \times 49}{216 \times 16 \times 81} = x $$

### Concept & Formula This problem tests your ability to express composite numbers as powers of their prime factors and then apply the laws of exponents. The primary rule used here is the product rule for exponents: $$ a^m \times a^n = a^{m+n} $$ ### Step-by-Step Solution * **Step 1:** Identify the base factors for the numerator. We recognize $343$ as $7^3$ and $49$ as $7^2$. Numerator $= 7^3 \times 7^2 = 7^{3+2} = 7^5$. * **Step 2:** Identify the base factors for the denominator. We recognize $216$ as $6^3$. Next, break down $16$ and $81$ into prime bases: $16 = 2^4$ and $81 = 3^4$. Combine these using exponent rules: $2^4 \times 3^4 = (2 \times 3)^4 = 6^4$. * **Step 3:** Combine the terms in the denominator. Denominator $= 6^3 \times 6^4 = 6^{3+4} = 6^7$. * **Step 4:** Construct the final fraction. Result $= \frac{7^5}{6^7}$. ### Exam Strategy & Shortcut Instead of converting everything to absolute prime bases (like $2$ and $3$), look at the answer options first. Notice that all options use base $6$ in the denominator. This gives you a massive clue to target base $6$ directly. Knowing your basic cubes ($6^3 = 216$) and recognizing that $16 \times 81 = 2^4 \times 3^4 = 6^4$ allows you to skip tedious prime factorization. ### Common Pitfall A common mistake is trying to multiply out the large numbers (e.g., $343 \times 49 = 16807$) before simplifying. This consumes massive amounts of time and increases the risk of arithmetic errors. Always factorize into powers first. ### Final Answer **Therefore, the correct answer is \frac{7^5}{6^7}.**