Let's solve the question step by step to find the simplified base of the function [tex]\( f(x) = \frac{1}{4} \left(\sqrt[3]{108}\right)^x \)[/tex].
1. Expression of [tex]\(f(x)\)[/tex]:
[tex]\[
f(x) = \frac{1}{4} \left(108^{\frac{1}{3}}\right)^x
\][/tex]
2. Simplify the base [tex]\( 108^{\frac{1}{3}} \)[/tex]:
We need to rewrite [tex]\( 108 \)[/tex] in terms of its prime factors.
[tex]\[
108 = 2^2 \times 3^3
\][/tex]
So,
[tex]\[
108^{\frac{1}{3}} = \left(2^2 \times 3^3\right)^{\frac{1}{3}}
\][/tex]
3. Distribute the cube root:
Apply the property of exponents:
[tex]\[
108^{\frac{1}{3}} = 2^{\frac{2}{3}} \times 3^{\frac{3}{3}}
\][/tex]
4. Simplify the exponents:
[tex]\[
3^{\frac{3}{3}} = 3^1 = 3
\][/tex]
[tex]\[
2^{\frac{2}{3}} = \sqrt[3]{4}
\][/tex]
5. Combine terms:
[tex]\[
108^{\frac{1}{3}} = 3 \times \sqrt[3]{4}
\][/tex]
Therefore, the simplified base of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
3 \sqrt[3]{4}
\][/tex]
Thus, the correct answer is:
[tex]\[
3 \sqrt[3]{4}
\][/tex]
