What is the simplified base for the function $f(x)=2\left(\sqrt[3]{27}^{2 x}\right)$?
Answer (2)
Simplify the cube root: 3 27 = 3 .
Rewrite the function: f ( x ) = 2 ( 3 2 x ) .
Apply the power of a power rule: f ( x ) = 2 (( 3 2 ) x ) .
Simplify to find the base: f ( x ) = 2 ( 9 x ) , so the base is 9 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 \t \t \t \t ( 3 27 2 x ) and asked to find the simplified base. This means we want to rewrite the function in the form f ( x ) = 2 ( b x ) where b is the simplified base.
Simplifying the Cube Root First, we simplify the expression inside the parenthesis. We know that 3 27 = 3 , since 3 × 3 × 3 = 27 .
Rewriting the Function Now we can rewrite the function as f ( x ) = 2 ( 3 2 x ) .
Applying the Power of a Power Rule Using the power of a power rule, we can rewrite 3 2 x as ( 3 2 ) x . So, f ( x ) = 2 (( 3 2 ) x ) .
Simplifying the Base Since 3 2 = 9 , we can further simplify the function to f ( x ) = 2 ( 9 x ) .
Final Answer Therefore, the simplified base for the function f ( x ) = 2 ( 3 27 2 x ) is 9.
Examples
Understanding exponential functions and how to simplify them is crucial in many real-world applications. For example, calculating compound interest involves exponential growth. If you invest money in an account that earns interest, the amount of money you have grows exponentially over time. Similarly, population growth, radioactive decay, and the spread of diseases can all be modeled using exponential functions. Simplifying the base of an exponential function helps in understanding the rate of growth or decay in these scenarios.
The simplified base for the function f ( x ) = 2 ( 3 27 2 x ) is 9 . This is found by simplifying 3 27 to 3 and rewriting the function accordingly. Thus, f ( x ) = 2 ( 9 x ) .
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