Let us simplify the base of the function [tex]\( f(x) = \frac{1}{4} (\sqrt[3]{108})^x \)[/tex].
We'll take it step by step:
1. Simplify [tex]\( \sqrt[3]{108} \)[/tex]:
[tex]\[
\sqrt[3]{108} = 108^{1/3}
\][/tex]
2. Factorize 108:
[tex]\[
108 = 4 \times 27 \quad \text{since} \quad 4 \times 27 = 108
\][/tex]
3. Express 108 as a product of two cube roots:
[tex]\[
108^{1/3} = (4 \times 27)^{1/3} = (4^{1/3} \times 27^{1/3})
\][/tex]
4. Simplify [tex]\( \sqrt[3]{27} \)[/tex]:
[tex]\[
27^{1/3} = 3 \quad \text{since} \quad 3^3 = 27
\][/tex]
5. Substitute [tex]\( 27^{1/3} \)[/tex] back in:
[tex]\[
108^{1/3} = 4^{1/3} \times 3
\][/tex]
6. Rewrite [tex]\( f(x) \)[/tex] using this result:
[tex]\[
f(x) = \frac{1}{4} (4^{1/3} \times 3)^x
\][/tex]
7. Separate the exponential terms:
[tex]\[
f(x) = \frac{1}{4} (4^{1/3})^x \times 3^x
\][/tex]
8. Combine the fraction with the simplified expression:
[tex]\[
= \frac{1}{4} \times (4^{x/3}) \times 3^x
\][/tex]
9. Simplify the base expression [tex]\( \frac{1}{4} \times 4^{x/3} \)[/tex]:
[tex]\[
\frac{1}{4} \times 4^{x/3} = \frac{4^{x/3}}{4} = 4^{x/3 - 1}
\][/tex]
Considering the base itself without an exponent:
10. Simplify base only:
[tex]\[
= \frac{4^{1/3} \times 3}{4}
= 4^{1/3} \times \frac{3}{4}
\][/tex]
11. Rewrite using 4 and will simplify gives us:
[tex]\[
= \frac{3 \cdot 4^{1/3}}{4}
\][/tex]
12. Combine:
[tex]\[
= \frac{3}{4^{2/3}}=3*4^{-2/3}
\][/tex]
13. Calculate numerical base simplification to approximate:
[tex]\[
\approx \boxed{1.1905507889761495}
\][/tex]
This simplified base of the function [tex]\( f(x)= \frac{1}{4}(\sqrt[3]{108})^x\approx 1.1905507889761495\)[/tex].
