To evaluate the expression \(\frac{(81^{\frac{3}{4}}-27^{\frac{1}{3}})}{3 \times 2^3}\), we will break it down step by step. ### Step 1: Simplify the Numerator **Calculate \(81^{\frac{3}{4}}\)**: - First, we recognize that \(81\) can be expressed as \(3^4\) because \(3^4 = 81\). - Therefore, we can rewrite \(81^{\frac{3}{4}}\) as: \[ (3^4)^{\frac{3}{4}} = 3^{4 \times \frac{3}{4}} = 3^3 = 27 \] **Calculate \(27^{\frac{1}{3}}\)**: - Similarly, \(27\) can be expressed as \(3^3\) because \(3^3 = 27\). - Thus, we can rewrite \(27^{\frac{1}{3}}\) as: \[ (3^3)^{\frac{1}{3}} = 3^{3 \times \frac{1}{3}} = 3^1 = 3 \] ### Step 2: Substitute Back into the Expression Now we substitute these results back into the numerator: \[ 81^{\frac{3}{4}} - 27^{\frac{1}{3}} = 27 - 3 = 24 \] ### Step 3: Simplify the Denominator **Calculate \(3 \times 2^3\)**: - First, calculate \(2^3\): \[ 2^3 = 8 \] - Now, multiply by \(3\): \[ 3 \times 8 = 24 \] ### Step 4: Combine the Results Now we can substitute the simplified numerator and denominator back into the expression: \[ \frac{(81^{\frac{3}{4}} - 27^{\frac{1}{3}})}{3 \times 2^3} = \frac{24}{24} = 1 \] ### Conclusion The final answer is \(1\). ### Explanation of Options - **Option A: 3** - This is incorrect because the calculations show that the numerator and denominator both simplify to \(24\), leading to a result of \(1\), not \(3\). - **Option B: 1** - This is correct as we have shown through our calculations that the expression evaluates to \(1\). - **Option C: \(\frac{1}{3}\)** - This is incorrect because the calculations do not support this value; the result is \(1\). - **Option D: \(\frac{1}{8}\)** - This is also incorrect as it does not match the evaluated result of \(1\). ### Revision Summary - \(81^{\frac{3}{4}} = 27\) and \(27^{\frac{1}{3}} = 3\). - The numerator simplifies to \(24\) and the denominator also simplifies to \(24\). - The final result of the expression is \(1\). - The correct answer is option B: \(1\).