Answer:
Simplified base would be 3∛4
Step-by-step explanation:
Given exponential function,
[tex]f(x)=\frac{1}{4}(\sqrt[3]{108})^x[/tex]
Since, in an exponential function [tex]f(x)=ab^x[/tex]
b is called base.
∵ 108 = 2 × 2 × 3 × 3 × 3,
Or 108 = 4 × 3³,
[tex]\implies \sqrt[3]{108} = \sqrt[3]{4\times 3^3}[/tex]
[tex]\sqrt[3]{108}=\sqrt[3]{4}\times \sqrt[3](3^3)[/tex] ( Using [tex]\sqrt[n]{ab}=\sqrt[n]{a}\times \sqrt[n]{b}[/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] ( Using power of power property of exponent )
[tex]\sqrt[3]{108}=3\sqrt[3]{4}[/tex]
Hence, the required simplified base would be 3∛4
i.e. SECOND option is correct.
