What is the base of the exponent in the function [tex]$f(x)=3(\sqrt[3]{8})^{2 x}$[/tex] when the function is written using only rational numbers and is in simplest form?
Answer (2)
Simplify the cube root: 3 8 = 2 .
Substitute the simplified cube root into the function: f ( x ) = 3 ( 2 ) 2 x .
Rewrite the exponent: f ( x ) = 3 ( 2 2 ) x .
Simplify the base: f ( x ) = 3 ( 4 ) x . The base of the exponent is 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = 3 ( 3 8 ) 2 x and we want to find the base of the exponent when the function is written using only rational numbers and is in simplest form.
Simplifying the Cube Root First, we simplify the cube root of 8. Since 2 3 = 8 , we have 3 8 = 2 .
Substituting the Value Now, substitute this value back into the function: f ( x ) = 3 ( 2 ) 2 x .
Rewriting the Exponent Next, we rewrite the exponent using the property of exponents ( a b ) c = a b c . So, f ( x ) = 3 ( 2 2 ) x .
Simplifying the Base Simplify the base: 2 2 = 4 . Therefore, f ( x ) = 3 ( 4 ) x .
Identifying the Base The function is now in the form f ( x ) = 3 ( 4 ) x . The base of the exponent is 4, which is a rational number.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. In finance, understanding the base of an exponential function helps in calculating the future value of an investment. For example, if you invest money in an account that compounds annually at a rate of r , the amount of money you have after t years is given by A = P ( 1 + r ) t , where P is the principal amount. Here, ( 1 + r ) is the base of the exponential function.
The base of the exponent in the function f ( x ) = 3 ( 3 8 ) 2 x is determined to be 4 after simplifying the cube root and applying exponent rules.
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