What is the simplified base of the function [tex]f(x)=\frac{1}{4}(\sqrt[3]{108})^x [/tex]?
Answer (2)
Simplify the cube root: 3 108 = 3 3 4 .
Substitute the simplified cube root into the function: f ( x ) = 4 1 ( 3 3 4 ) x .
Identify the base of the exponential function: 4 3 3 4 .
The simplified base is 4 3 3 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 1 ( 3 108 ) x and asked to simplify the base of this exponential function.
Simplifying the Cube Root First, let's simplify the cube root of 108. We can write 108 as 108 = 27 × 4 = 3 3 × 4 . Therefore, 3 108 = 3 3 3 × 4 = 3 3 3 × 3 4 = 3 3 4 .
Substituting Back Now, substitute this back into the original function: f ( x ) = 4 1 ( 3 3 4 ) x .
Identifying the Base The base of the exponential function is therefore 4 1 ( 3 3 4 ) = 4 3 3 4 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Simplifying the base of an exponential function can make it easier to analyze and understand the behavior of the function. For example, in finance, understanding the base of an exponential function representing compound interest helps in calculating the effective interest rate and comparing different investment options. In biology, it helps in modeling population growth or decay under certain conditions.
The simplified base of the function f ( x ) = 4 1 ( 3 108 ) x is 4 3 3 4 . This is found by simplifying the cube root of 108 to get 3 3 4 and then factoring in the 4 1 .
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