Which exponential functions have been simplified correctly? Check all that apply.
[tex]f(x)=5 \sqrt[3]{16}^x=5(2 \sqrt[3]{2})^x[/tex]
[tex]f(x)=2.3(8)^{\frac{1}{2} x}=2.3(4)^x[/tex]
[tex]f(x)=81^{\frac{x}{4}}=3^x[/tex]
[tex]f(x)=\frac{3}{4} \sqrt{27}^x=\frac{3}{4}(3 \sqrt{3})^x[/tex]
[tex]f(x)=(24)^{\frac{1}{3} x}=2(\sqrt[3]{3})^x[/tex]
Answer (2)
Simplify each exponential function.
Compare the simplified form with the given simplified form.
Identify the functions that have been simplified correctly.
The functions simplified correctly are: f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x , f ( x ) = 8 1 4 x = 3 x , f ( x ) = 4 3 27 x = 4 3 ( 3 3 ) x .
Explanation
Understanding the Problem We are given 5 exponential functions and need to check which ones have been simplified correctly.
Listing the Functions The functions are:
f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x
f ( x ) = 2.3 ( 8 ) 2 1 x = 2.3 ( 4 ) x
f ( x ) = 8 1 4 x = 3 x
f ( x ) = 4 3 27 x = 4 3 ( 3 3 ) x
f ( x ) = ( 24 ) 3 1 x = 2 ( 3 3 ) x
Solution Plan We will simplify each of the given exponential functions and compare the simplified form with the given simplified form.
Simplifying Function 1
f ( x ) = 5 3 16 x = 5 ( 1 6 3 1 ) x = 5 (( 2 4 ) 3 1 ) x = 5 ( 2 3 4 ) x = 5 ( 2 ⋅ 2 3 1 ) x = 5 ( 2 3 2 ) x . So, the first function is simplified correctly.
Simplifying Function 2
f ( x ) = 2.3 ( 8 ) 2 1 x = 2.3 ( 8 2 1 ) x = 2.3 (( 2 3 ) 2 1 ) x = 2.3 ( 2 2 3 ) x = 2.3 ( 2 2 ) x e q 2.3 ( 4 ) x . So, the second function is not simplified correctly.
Simplifying Function 3
f ( x ) = 8 1 4 x = ( 8 1 4 1 ) x = ( 3 4 ) 4 x = 3 x . So, the third function is simplified correctly.
Simplifying Function 4
f ( x ) = 4 3 27 x = 4 3 ( 2 7 2 1 ) x = 4 3 (( 3 3 ) 2 1 ) x = 4 3 ( 3 2 3 ) x = 4 3 ( 3 3 ) x . So, the fourth function is simplified correctly.
Simplifying Function 5
f ( x ) = ( 24 ) 3 1 x = ( 2 4 3 1 ) x = (( 2 3 ⋅ 3 ) 3 1 ) x = ( 2 ⋅ 3 3 1 ) x = ( 2 3 3 ) x . So, the fifth function is simplified correctly.
Final Answer The exponential functions that have been simplified correctly are:
f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x f ( x ) = 8 1 4 x = 3 x f ( x ) = 4 3 27 x = 4 3 ( 3 3 ) x
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that earns compound interest, the amount of money in your account will grow exponentially over time. Understanding how to simplify exponential functions can help you make informed financial decisions.
The functions that have been simplified correctly are: 1. f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x , 2. f ( x ) = 8 1 4 x = 3 x , and 3. f ( x ) = 4 3 27 x = 4 3 ( 3 3 ) x . The second and fifth functions are not simplified correctly.
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