Question

What is the simplified base of the function f(x) = One-fourth (Root Index 3 StartRoot 108 EndRoot) Superscript x?

Answer 1

The base is:
`3 \sqrt[3]{4}`

Explanation:

Given

`f(x) = (1)/(4)(\sqrt[3]{108})^x`

Required

The base

Expand 108

`f(x) = (1)/(4)(\sqrt[3]{3^3 * 4})^x`

Rewrite the exponent as:

`f(x) = (1)/(4)(3^3 * 4)^(1)/(3)^x`

Expand

`f(x) = (1)/(4)(3^3^(1)/(3) * 4^(1)/(3))^x`

`f(x) = (1)/(4)(3 * 4^(1)/(3))^x`

Rewrite as:

`f(x) = (1)/(4)(3 \sqrt[3]{4})^x`

An exponential function has the following form:

`f(x)=ab^x`

Where

`b \to base`

By comparison:

`b =3 \sqrt[3]{4}`

So, the base is:
`3 \sqrt[3]{4}`