mplify the expression given below. \[ \left(81 d^{24}\right)^{-\frac{3}{4}} \] A. \( -\frac{d^{3}}{27} \) B. \( \frac{1}{2 \pi \varepsilon^{-x}} \) C. \( \frac{27}{d^{2}} \) D. \( -27 d^{23} \)

We wish to simplify \[ \left(81 d^{24}\right)^{-\frac{3}{4}}. \] Let’s go through the process step by step. --- ### Step 1. Express 81 as a power of 3 We have \[ 81 = 3^4. \] Thus, we can write the expression as \[ \left(3^4 d^{24}\right)^{-\frac{3}{4}}. \] --- ### Step 2. Distribute the exponent to each factor Recall the rule \[ (ab)^c = a^c b^c. \] Applying this rule gives \[ \left(3^4 d^{24}\right)^{-\frac{3}{4}} = \left(3^4\right)^{-\frac{3}{4}} \cdot \left(d^{24}\right)^{-\frac{3}{4}}. \] --- ### Step 3. Simplify each factor 1. For the first factor: \[ \left(3^4\right)^{-\frac{3}{4}} = 3^{4 \cdot \left(-\frac{3}{4}\right)} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}. \] 2. For the second factor: \[ \left(d^{24}\right)^{-\frac{3}{4}} = d^{24 \cdot \left(-\frac{3}{4}\right)} = d^{-18}. \] --- ### Step 4. Combine the results The simplified expression is \[ \frac{1}{27} \cdot d^{-18} = \frac{1}{27\, d^{18}}. \] --- ### Conclusion The completely simplified form of \[ \left(81 d^{24}\right)^{-\frac{3}{4}} \] is \[ \frac{1}{27\, d^{18}}. \] --- ### Comparison with Given Options The provided answer choices are: A. \( -\frac{d^{3}}{27} \) B. \( \frac{1}{2 \pi \varepsilon^{-x}} \) C. \( \frac{27}{d^{2}} \) D. \( -27 d^{23} \) None of these match the correct simplified form \(\frac{1}{27\, d^{18}}\). --- ### Final Answer The correct simplified expression is \[ \frac{1}{27\, d^{18}}, \] which does not correspond to any of the given options.