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}^{\dot{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.
Check if the simplified form matches the given simplified form.
Identify the functions that have been simplified correctly.
The correctly simplified functions are: f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x , f ( x ) = 8 1 4 x = 3 x , and 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:
$f(x)=5
\sqrt[3]{16}^x=5(2
\sqrt[3]{2})^x 2. f(x)=2.3(8)^{\frac{1}{2} x}=2.3(4)^x 3. f(x)=81^{\frac{x}{4}}=3^x 4. f(x)=
\frac{3}{4}
\sqrt{27}^{\dot{x}}=
\frac{3}{4}(3
\sqrt{3})^x 5. f(x)=(24)^{\frac{1}{3} x}=2(\sqrt[3]{3})^x$
Analyzing Function 1 Let's analyze each function:
f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x . We can simplify 3 16 as 3 2 4 = 2 4/3 = 2 ⋅ 2 1/3 = 2 3 2 . Therefore, f ( x ) = 5 ( 2 3 2 ) x . This simplification is correct.
Analyzing Function 2
f ( x ) = 2.3 ( 8 ) 2 1 x = 2.3 ( 4 ) x . We can rewrite 8 2 1 x as ( 8 2 1 ) x = ( 8 ) x = ( 2 2 ) x . Thus, f ( x ) = 2.3 ( 2 2 ) x . Since 2.3 ( 4 ) x = 2.3 ( 2 2 ) x = 2.3 ( 2 2 x ) , the original simplification is incorrect.
Analyzing Function 3
f ( x ) = 8 1 4 x = 3 x . We can rewrite 81 as 3 4 . Therefore, f ( x ) = ( 3 4 ) 4 x = 3 4 ⋅ 4 x = 3 x . This simplification is correct.
Analyzing Function 4
$f(x)=
\frac{3}{4}
\sqrt{27}^{\dot{x}}=
\frac{3}{4}(3
\sqrt{3})^x . W ec an s im pl i f y \sqrt{27} a s \sqrt{3^3} = 3^{3/2} = 3 \sqrt{3} . T h ere f ore , f(x) =
\frac{3}{4} (3 \sqrt{3})^x$. This simplification is correct.
Analyzing Function 5
f ( x ) = ( 24 ) 3 1 x = 2 ( 3 3 ) x . We can rewrite 24 as 2 3 ⋅ 3 . Therefore, f ( x ) = ( 2 3 ⋅ 3 ) 3 1 x = ( 2 3 ) 3 1 x ⋅ 3 3 1 x = 2 x ( 3 3 ) x = ( 2 3 3 ) x . The original simplification is incorrect.
Final Answer In summary, the correctly simplified functions are:
f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x f ( x ) = 8 1 4 x = 3 x $f(x)=
\frac{3}{4}
\sqrt{27}^{\dot{x}}=
\frac{3}{4}(3
\sqrt{3})^x$
Examples
Exponential functions are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For instance, if you invest money in a bank account with compound interest, the amount of money you have will grow exponentially over time. Understanding how to simplify exponential functions can help you make informed decisions about investments and other financial matters.
The correctly simplified functions are 1, 3, and 4. They are f ( x ) = 5 3 16 x = 5 ( 2 3 2 ) x , f ( x ) = 8 1 4 x = 3 x , and f ( x ) = 4 3 27 x ˙ = 4 3 ( 3 3 ) x . Functions 2 and 5 contain errors in their simplifications.
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