$(3)^8 \times (3)^4 = x$

## Concept & Formula This question requires applying the product rule of indices first, and then converting the resulting base into a larger, equivalent exponential form to match the options. $$ a^m \times a^n = a^{m + n} $$ $$ a^{m \times n} = (a^m)^n $$ ## Step-by-Step Solution * **Given:** The expression is $(3)^8 \times (3)^4$. * **Calculation / Deduction:** * First, apply the product rule since the bases are identical: $$ 3^{8 + 4} = 3^{12} $$ * Our simplified result is $3^{12}$. Now we must test the options to find an equivalent value. * Let's test the base $27$ options: We know $27 = 3^3$. * Option (a): $(27)^3 = (3^3)^3 = 3^9$. (Incorrect) * Option (b): $(27)^5 = (3^3)^5 = 3^{15}$. (Incorrect) * Let's test the base $729$ options: We know $729 = 3^6$. * Option (c): $(729)^2 = (3^6)^2 = 3^{12}$. (Matches exactly!) * Option (d): $(729)^3 = (3^6)^3 = 3^{18}$. (Incorrect) ## Exam Strategy & Shortcut Once you calculate $3^{12}$, quickly scan the bases in the options: $27$ and $729$. Recognize that $27 = 3^3$ and $729 = 3^6$. To get a power of $12$, you need either $(3^3)^4$ or $(3^6)^2$. The option $(729)^2$ perfectly represents $(3^6)^2$. ## Common Pitfall A very common mistake is finding $3^{12}$, not seeing it immediately in the options, and impulsively selecting "None of these". Exam creators intentionally format the correct answer with a different base to ensure you truly understand how exponent manipulation works in both directions. ## Final Answer **Therefore, the correct answer is (729)^2.**