Q:

What is the simplified base of the function f(x) = One-fourth (Root Index 3 StartRoot 108 EndRoot) Superscript x?33RootIndex 3 StartRoot 4 EndRoot6RootIndex 3 StartRoot 3 EndRoot27Answer: second choice

Accepted Solution

A:
Answer:Simplified base would be 3∛4Step-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∛4i.e. SECOND option is correct.