Answer:
The required simplified base would be 3∛4
Step-by-step explanation:
Given exponential function that:
[tex]f(x)=\frac{1}{4}(\sqrt[3]{108})^x[/tex]
As we can see, 108 is the base of the exponential function with the form:
f(x) = a[tex]b^{x}[/tex]
So, we can factor 108 = 2 × 2 × 3 × 3 × 3
<=> 108 = 4 × 3³
Hence, [tex]\sqrt[3]{108} = \sqrt[3]{4\times 3^3}[/tex]
So we have:
[tex]\sqrt[3]{108}=\sqrt[3]{4}\times \sqrt[3](3^3)[/tex]
<=> [tex]\sqrt[3]{108}=\sqrt[3]{4}\times (3^3)^\frac{1}{3}[/tex]
<=> [tex]\sqrt[3]{108}=\sqrt[3]{4}\times 3^{3\times \frac{1}{3}}[/tex]
<=> [tex]\sqrt[3]{108}=3\sqrt[3]{4}[/tex]
Hence, the required simplified base would be 3∛4
