D. 12. \( [2002 / 1] \) Evaluate \( \frac{(81)^{3 / 4}-(27)^{1 / 3}}{3 \times 2^{3}} \) A. 1. B. \( \frac{1}{8} \) C. 3. D. \( \frac{1}{3} \cdot[2005 / 7] \)

To evaluate the expression \[ \frac{(81)^{3 / 4}-(27)^{1 / 3}}{3 \times 2^{3}}, \] let's break it down step by step. ### Step 1: Calculate \( (81)^{3/4} \) First, we can express \( 81 \) as \( 3^4 \): \[ (81)^{3/4} = (3^4)^{3/4} = 3^{4 \cdot (3/4)} = 3^3 = 27. \] ### Step 2: Calculate \( (27)^{1/3} \) Next, we express \( 27 \) as \( 3^3 \): \[ (27)^{1/3} = (3^3)^{1/3} = 3^{3 \cdot (1/3)} = 3^1 = 3. \] ### Step 3: Substitute back into the expression Now we substitute these results back into the original expression: \[ \frac{(81)^{3/4} - (27)^{1/3}}{3 \times 2^{3}} = \frac{27 - 3}{3 \times 2^{3}}. \] ### Step 4: Simplify the numerator Calculating the numerator: \[ 27 - 3 = 24. \] ### Step 5: Calculate the denominator Now, calculate the denominator: \[ 3 \times 2^{3} = 3 \times 8 = 24. \] ### Step 6: Final calculation Now we can simplify the entire expression: \[ \frac{24}{24} = 1. \] ### Conclusion The final result is \[ \boxed{1}. \] Thus, the answer is A. 1.