The value of $\left( \frac{9^2 \times 18^4}{3^{16}} \right)$ is

## Concept & Formula This problem requires decomposing composite bases into their prime factors to simplify a complex fractional exponential expression. The primary index laws used are: $$(a \times b)^n = a^n \times b^n$$ $$(a^m)^n = a^{m \times n}$$ $$\frac{a^m}{a^n} = a^{m - n}$$ ## Step-by-Step Solution * **Given:** The expression to evaluate is $\frac{9^2 \times 18^4}{3^{16}}$. * **Calculation / Deduction:** * Step 1: Prime factorize all the bases in the numerator. * $9 = 3^2$ * $18 = 2 \times 9 = 2 \times 3^2$ * Step 2: Substitute these prime factorizations back into the expression. $$\frac{(3^2)^2 \times (2 \times 3^2)^4}{3^{16}}$$ * Step 3: Apply the power rules to remove the brackets. * $(3^2)^2 = 3^4$ * $(2 \times 3^2)^4 = 2^4 \times (3^2)^4 = 2^4 \times 3^8$ * The numerator becomes: $3^4 \times 2^4 \times 3^8$ * Step 4: Combine the terms with the same base in the numerator. * $2^4 \times (3^4 \times 3^8) = 2^4 \times 3^{12}$ * Step 5: Substitute the simplified numerator back over the denominator. $$\frac{2^4 \times 3^{12}}{3^{16}}$$ * Step 6: Apply the division rule of indices for base $3$. $$2^4 \times 3^{12 - 16} = 2^4 \times 3^{-4}$$ * Step 7: Convert the negative exponent into a fraction and evaluate. $$\frac{2^4}{3^4} = \frac{16}{81}$$ ## Exam Strategy & Shortcut Break numbers down to prime bases immediately. Once you see $9$ and $18$, you know the entire expression consists of base $2$ and base $3$. Tally the powers of $3$ in the numerator mentally: $4$ (from $9^2$) + $8$ (from $18^4$) = $12$. Tally powers of $3$ in denominator: $16$. Net power of $3$ is $-4$ (meaning $3^4$ or $81$ is in the denominator). Tally powers of $2$ in numerator: $4$ (from $18^4$). Numerator is $2^4 = 16$. The answer is $\frac{16}{81}$. ## Common Pitfall A standard error is miscalculating the expansion of $(2 \times 3^2)^4$. Some students forget to distribute the power of $4$ to the $2$, resulting in an incorrect factor of $2$ instead of $16$. Always distribute the outer exponent to every single factor inside the parenthesis. ## Final Answer **Therefore, the correct answer is \frac{16}{81}.**