To simplify the expression \((\sqrt[3]{64a^{3}})^{-1}\), let's break it down step by step. ### Step 1: Simplify the Cube Root The expression inside the parentheses is \(\sqrt[3]{64a^{3}}\). We can simplify this by breaking it into two parts: the cube root of \(64\) and the cube root of \(a^{3}\). 1. **Calculate \(\sqrt[3]{64}\)**: - The number \(64\) can be expressed as \(4^3\) because \(4 \times 4 \times 4 = 64\). - Therefore, \(\sqrt[3]{64} = 4\). 2. **Calculate \(\sqrt[3]{a^{3}}\)**: - The cube root of \(a^{3}\) is simply \(a\) because \((a)^{3} = a^{3}\). - Therefore, \(\sqrt[3]{a^{3}} = a\). Combining these results, we have: \[ \sqrt[3]{64a^{3}} = \sqrt[3]{64} \cdot \sqrt[3]{a^{3}} = 4a. \] ### Step 2: Apply the Negative Exponent Now we need to apply the negative exponent from the original expression: \[ (\sqrt[3]{64a^{3}})^{-1} = (4a)^{-1}. \] Using the property of exponents that states \(x^{-1} = \frac{1}{x}\), we can rewrite this as: \[ (4a)^{-1} = \frac{1}{4a}. \] ### Final Answer Thus, the simplified form of \((\sqrt[3]{64a^{3}})^{-1}\) is: \[ \frac{1}{4a}. \] ### Correct Option The correct option is **D. \( \frac{1}{4a} \)**. ### Explanation of Other Options - **Option A: \(4a\)**: This option is incorrect because it does not account for the negative exponent. The expression simplifies to \(\frac{1}{4a}\), not \(4a\). - **Option B: \(\frac{1}{8a}\)**: This option is incorrect because it miscalculates the cube root of \(64\). The correct cube root is \(4\), not \(8\). - **Option C: \(8a\)**: This option is also incorrect as it fails to apply the negative exponent correctly. The expression simplifies to \(\frac{1}{4a}\), not \(8a\). ### Summary of Key Points - The cube root of \(64\) is \(4\) and the cube root of \(a^{3}\) is \(a\). - The expression simplifies to \(4a\) before applying the negative exponent. - Applying the negative exponent gives \(\frac{1}{4a}\). - The correct answer is option D: \(\frac{1}{4a}\). This thorough breakdown should help you understand how to simplify similar expressions in the future!