To determine the simplified base of the function [tex]\( f(x) = \frac{1}{4}(\sqrt[3]{108})^* \)[/tex], we need to first simplify the expression inside the cubic root, [tex]\(\sqrt[3]{108}\)[/tex].
1. Prime Factorization of 108:
[tex]\[
108 = 2^2 \times 3^3
\][/tex]
2. Applying Cube Root:
[tex]\[
\sqrt[3]{108} = \sqrt[3]{2^2 \times 3^3}
\][/tex]
3. Distribute the Cube Root to Prime Factors:
[tex]\[
\sqrt[3]{2^2 \times 3^3} = \sqrt[3]{2^2} \times \sqrt[3]{3^3}
\][/tex]
4. Simplify Each Term:
[tex]\[
\sqrt[3]{3^3} = 3 \quad \text{(since the cubic root of } 3^3 \text{ is } 3\text{)}
\][/tex]
[tex]\[
\sqrt[3]{2^2} = 2^{2/3}
\][/tex]
5. Combine the Simplified Terms:
[tex]\[
\sqrt[3]{108} = 3 \times 2^{2/3}
\][/tex]
To express [tex]\( 2^{2/3} \)[/tex] in a more readable form, we can rephrase it as [tex]\( \sqrt[3]{4} \)[/tex] because:
[tex]\[
2^{2/3} = \sqrt[3]{2^2} = \sqrt[3]{4}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[3]{108}\)[/tex] is:
[tex]\[
3 \sqrt[3]{4}
\][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = \frac{1}{4}(\sqrt[3]{108})^* \)[/tex] is [tex]\( 3\sqrt[3]{4} \)[/tex].
So, the correct answer is:
[tex]\[
3\sqrt[3]{4}
\][/tex]
