What is the simplified base of the function \( f(x)=\frac{1}{4}(\sqrt[3]{108})^{x} \) ? 3 \( 3 \sqrt[3]{4} \) \( 6 \sqrt[3]{3} \) 27

To simplify the function \( f(x)=\frac{1}{4}(\sqrt[3]{108})^{x} \), we start by simplifying \( \sqrt[3]{108} \). First, break down \( 108 \) into its prime factors: \( 108 = 2^2 \cdot 3^3 \). Taking the cube root gives us: \[ \sqrt[3]{108} = \sqrt[3]{2^2 \cdot 3^3} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} = \sqrt[3]{4} \cdot 3. \] Now substituting this back into the function: \[ f(x) = \frac{1}{4}(3\sqrt[3]{4})^{x} = \frac{1}{4} \cdot 3^{x} \cdot (\sqrt[3]{4})^{x} = \frac{3^{x}}{4} \cdot (\sqrt[3]{4})^{x}. \] Since \( \frac{3^{x}}{4} \) can also be simplified to maintain a common base, we see that \( 3\sqrt[3]{4} \) is indeed a valid base choice for simplification. Hence, the simplified base can be expressed functionally in terms of \( x \). To confirm further clarity, if we analyze common mistakes here, remember that changing the base without correctly accounting for the manipulation could lead to confusion about how various components interact, especially when dealing with roots and exponents.