Simplify the radical expression. The variable represents a positive real number. \[ 2 \sqrt[3]{-81 a^{3}} \]

To simplify the expression \( 2 \sqrt[3]{-81 a^{3}} \), first, break it down into its components. The cube root of \(-81\) can be calculated as follows: \(-81\) can be expressed as \(-1 \cdot 81\) or \(-1 \cdot 3^4\). The cube root of \(-1\) is \(-1\), and the cube root of \(81\) can be computed as \(\sqrt[3]{3^4} = 3^{4/3} = 3^{1 + 1/3} = 3 \cdot \sqrt[3]{3}\). Next, we handle \( a^3 \): the cube root of \( a^3 \) is simply \( a \). Putting all this together, we have: \[ \sqrt[3]{-81 a^{3}} = \sqrt[3]{-1} \cdot \sqrt[3]{81} \cdot \sqrt[3]{a^{3}} = -3 \cdot \sqrt[3]{3} \cdot a \] Now, multiplying by the 2 at the front gives: \[ 2 \sqrt[3]{-81 a^{3}} = 2 \cdot (-3) \cdot \sqrt[3]{3} \cdot a = -6a \sqrt[3]{3} \] So, the simplified expression is: \[ -6a \sqrt[3]{3} \]