To determine the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex], let's proceed step-by-step:
1. Calculate the cube root of 108:
[tex]\[
\sqrt[3]{108} \approx 4.762203155904598
\][/tex]
2. Express 108 as a product of prime factors:
[tex]\[
108 = 2^2 \cdot 3^3
\][/tex]
3. Find the cube root of 108 in terms of its prime factors:
[tex]\[
\sqrt[3]{108} = \sqrt[3]{2^2 \cdot 3^3}
\][/tex]
This can be broken down as:
[tex]\[
\sqrt[3]{2^2} = 2^{2/3}
\][/tex]
and
[tex]\[
\sqrt[3]{3^3} = 3
\][/tex]
Therefore,
[tex]\[
\sqrt[3]{108} = 2^{2/3} \cdot 3
\][/tex]
4. Simplify the base of the function:
[tex]\[
\text{Base} = \frac{1}{4} \sqrt[3]{108}
\][/tex]
Substitute [tex]\(\sqrt[3]{108}\)[/tex] with its simplified form:
[tex]\[
\text{Base} = \frac{1}{4} (2^{2/3} \cdot 3)
\][/tex]
5. Combine and simplify the expression:
[tex]\[
\text{Base} = \frac{1}{4} \cdot 2^{2/3} \cdot 3
\][/tex]
Since [tex]\(\frac{1}{4} = 2^{-2}\)[/tex], we can write:
[tex]\[
\text{Base} = 2^{-2} \cdot 2^{2/3} \cdot 3 = 2^{2/3 - 2} \cdot 3
\][/tex]
Simplifying [tex]\(2^{2/3 - 2}\)[/tex]:
[tex]\[
2^{2/3 - 2} = 2^{2/3 - 6/3} = 2^{-4/3}
\][/tex]
So,
[tex]\[
\text{Base} = 2^{-4/3} \cdot 3
\][/tex]
6. Converting the base to a decimal form:
Using the numerical result, we have:
[tex]\[
2^{-4/3} \cdot 3 \approx 0.520020955762976
\][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex] is approximately [tex]\( 0.520020955762976 \)[/tex]. This results from a combination of the cube root of 108 and the fraction [tex]\(\frac{1}{4}\)[/tex].
