To find the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex], let's delve into the details.
First, we need to simplify the expression inside the cube root: [tex]\(\sqrt[3]{108}\)[/tex].
Note that [tex]\(108\)[/tex] can be decomposed into factors:
[tex]\[ 108 = 27 \times 4 \][/tex]
[tex]\[ 27 = 3^3 \][/tex]
Thus,
[tex]\[ 108 = 3^3 \times 4 \][/tex]
Taking the cube root of both sides, we get:
[tex]\[ \sqrt[3]{108} = \sqrt[3]{3^3 \times 4} \][/tex]
[tex]\[ \sqrt[3]{108} = \sqrt[3]{3^3} \times \sqrt[3]{4} \][/tex]
[tex]\[ \sqrt[3]{108} = 3 \times \sqrt[3]{4} \][/tex]
So, the simplified base can be written as:
[tex]\[ 3 \sqrt[3]{4} \][/tex]
Therefore, the valid answer among the given options is:
[tex]\[ 3 \sqrt[3]{4} \][/tex]
