To solve the problem of finding the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex], let's proceed step-by-step.
1. Identify the constant factor and the radicand:
The function has a constant factor of [tex]\(\frac{1}{4}\)[/tex] and a base that involves the cube root of [tex]\(108\)[/tex].
2. Find the cube root of 108:
Recall that the cube root of [tex]\(108\)[/tex] can be represented as:
[tex]\[
\sqrt[3]{108}
\][/tex]
3. Combine the constant factor with the cube root of 108:
We combine the constant [tex]\(\frac{1}{4}\)[/tex] with the cube root of 108 to find the base of the function. The expression becomes:
[tex]\[
\frac{1}{4} \sqrt[3]{108}
\][/tex]
4. Simplify the expression if possible:
To better understand the simplified form of [tex]\(\frac{1}{4} \sqrt[3]{108}\)[/tex], first note that [tex]\(108\)[/tex] can be factored into:
[tex]\[
108 = 4 \times 27 = 4 \times 3^3
\][/tex]
5. Extract the cube root:
Using the factorization, we get:
[tex]\[
\sqrt[3]{108} = \sqrt[3]{4 \times 3^3}
\][/tex]
Since [tex]\(\sqrt[3]{3^3} = 3\)[/tex], the expression simplifies to:
[tex]\[
\sqrt[3]{108} = 3 \sqrt[3]{4}
\][/tex]
6. Substitute back into the expression:
Now substitute [tex]\(\sqrt[3]{108}\)[/tex] with [tex]\(3\sqrt[3]{4}\)[/tex] in the original simplified base expression:
[tex]\[
\frac{1}{4} \sqrt[3]{108} = \frac{1}{4} (3 \sqrt[3]{4})
\][/tex]
7. Simplify further:
We can distribute [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \times 3 \sqrt[3]{4} = \frac{3}{4} \sqrt[3]{4}
\][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex] is:
[tex]\[
\boxed{3 \sqrt[3]{4}}
\][/tex]
