To determine the simplified base of the given function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex], let's go step-by-step:
1. Understand the original function: We are provided with [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex].
2. Simplify the cube root of 108:
- The cube root of 108 is a real number. Numerically, this value is approximately [tex]\( 4.762 \)[/tex].
3. Determine the base:
- We multiply this cube root by [tex]\(\frac{1}{4}\)[/tex]. Numerically, calculating [tex]\(\frac{1}{4} \times 4.762 = 1.191\)[/tex].
Thus, the base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex] is approximately [tex]\(1.191\)[/tex]. Therefore, none of the answer choices given [tex]\( (3, 3 \sqrt[3]{4}, 6 \sqrt[3]{3}, 27) \)[/tex] match the simplified numerical base [tex]\( 1.191 \)[/tex].
However, to consider the base in terms of one of the provided forms:
- Rewrite [tex]\(108\)[/tex] as [tex]\(108 = 27 \times 4 = 3^3 \times 4\)[/tex].
- Hence, [tex]\( \sqrt[3]{108} = \sqrt[3]{3^3 \times 4} \)[/tex].
- Using the property of exponents, [tex]\(\sqrt[3]{3^3 \times 4} = 3 \times \sqrt[3]{4} \)[/tex].
Therefore, the simplified base in the given forms is [tex]\( \frac{1}{4} \times 3 \sqrt[3]{4}\)[/tex], which exactly matches one of your given choices.
So, the answer is:
[tex]\[
3 \sqrt[3]{4}
\][/tex]
