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

## Concept & Strategy When dealing with a massive fraction of composite numbers, immediately convert every number into its prime or fundamental exponential base. This transforms a difficult arithmetic problem into a simple index addition problem. ## Step-by-Step Solution * **Given:** The fraction is $\frac{343 \times 49}{216 \times 16 \times 81}$. * **Calculation / Deduction:** * Step 1: Convert the numbers in the numerator to their base powers. * $343 = 7^3$ * $49 = 7^2$ * The numerator becomes: $7^3 \times 7^2 = 7^{3 + 2} = 7^5$ * Step 2: Convert the numbers in the denominator to their base powers. Look for patterns related to base $6$, as the options suggest it. * $216 = 6^3$ * For the remaining terms, factorize them: * $16 = 2^4$ * $81 = 3^4$ * Combine $16 \times 81 = 2^4 \times 3^4$. Because the exponents are the same, we can group the bases: $(2 \times 3)^4 = 6^4$. * The denominator becomes: $6^3 \times 6^4 = 6^{3 + 4} = 6^7$ * Step 3: Combine the simplified numerator and denominator. $$ \frac{7^5}{6^7} $$ * Checking the options, this matches Option (a) perfectly. ## Exam Strategy & Shortcut Use the options as a cheat sheet. The options show numerators of base $7$ and denominators of base $6$. This tells you exactly what bases to target. You only need to verify the powers. Numerator: $343 \times 49$ is obviously $7^3 \times 7^2 = 7^5$. Denominator: $216$ is $6^3$. The rest is $16 \times 81 = (2 \times 3)^4 = 6^4$. Total power of $6$ is $3+4=7$. The answer is instantly $\frac{7^5}{6^7}$. ## Common Pitfall The worst approach you can take is trying to multiply out the numerator ($16807$) and denominator ($279936$) to create a massive fraction, and then attempting to simplify it. Always look at the format of the options before you start calculating; they dictate the most efficient path. ## Final Answer **Therefore, the correct answer is \frac{7^5}{6^7}.**