$(19)^{12} \times (19)^8 \div (19)^4 = (19)^x$

## Concept & Formula This problem tests the direct application of the product and quotient rules of indices for a uniform base. Since the base is identical across all terms, we only need to perform basic arithmetic on the exponents. $$ a^m \times a^n = a^{m + n} $$ $$ \frac{a^m}{a^n} = a^{m - n} $$ ## Step-by-Step Solution * **Given:** The equation is $(19)^{12} \times (19)^8 \div (19)^4 = (19)^x$. * **Calculation / Deduction:** * Since all bases are exactly the same ($19$), we can combine the operations into a single exponent equation. * For multiplication, add the exponents: $12 + 8$. * For division, subtract the exponent: $- 4$. * Combine them on the left side: $$ (19)^{12 + 8 - 4} = (19)^x $$ $$ (19)^{20 - 4} = (19)^x $$ $$ (19)^{16} = (19)^x $$ * By equating the powers of the identical bases, we find: $$ x = 16 $$ * Reviewing the provided options: (a) $6$, (b) $8$, (c) $12$, (d) $24$. The correct answer $16$ is not listed. ## Exam Strategy & Shortcut You do not need to rewrite the bases. Simply isolate the exponents visually and calculate mentally: $12 + 8 = 20$. Then $20 - 4 = 16$. Scan the options for $16$. If it is absent, immediately confidently select "None of these". This should take roughly 3 seconds. ## Common Pitfall A common mistake is misinterpreting the order of operations or the exponent rules, such as multiplying the exponents instead of adding them, or getting confused and applying division to the base numbers themselves. Stick strictly to the index laws. ## Final Answer **Therefore, the correct answer is None of these.**