Answer:
Option A, C, D, are the correct options.
Step-by-step explanation:
To check the exponential functions which have been simplified correctly we will take option one by one.
A). [tex]f(x)=5(\sqrt[3]{16})^{x}=5(\sqrt[3]{2^{4}})^{x}=5(2\sqrt[3]{2})^{x}[/tex]
B). [tex]f(x)=2.3(8^{\frac{1}{2}})^{x}=2.3(2.\sqrt{2})^{x}\neq 2.3(4^{x})[/tex]
C). [tex]f(x)=81^{\frac{x}{4}}=(3^{4})^{\frac{x}{4}}=(3^{\frac{4}{4}})^{x}=3^{x}[/tex]
D). [tex]f(x)=\frac{3}{4}(\sqrt{27})^{x}=\frac{3}{4}(\sqrt{3^{3}})^{x}=\frac{3}{4}.(3\sqrt{3})^{x}[/tex]
[tex]f(x)=(24)^{\frac{1}{3}x}=(2^{3}.3)^{\frac{1}{3}x}=(2.3^{\frac{1}{3}})^{x}\neq 2(\sqrt[3]{3})^{x}[/tex]
The given exponential function which are correctly simplified are:[tex]f(x) = 5(\sqrt[3]{16})^x[/tex], [tex]f(x) = 2.3(8)^{\frac{1}{2}x}[/tex], [tex]f(x) = (81)^{\frac{x}{4}}[/tex], and [tex]f(x)= \dfrac{3}{4}(\sqrt{27})^x[/tex] and this can be determine by usnig the arithmetic operations.
Given :
Exponential Functions -
1). [tex]f(x) = 5(\sqrt[3]{16})^x[/tex]
2). [tex]f(x) = 2.3(8)^{\frac{1}{2}x}[/tex]
3). [tex]f(x) = (81)^{\frac{x}{4}}[/tex]
4). [tex]f(x)= \dfrac{3}{4}(\sqrt{27})^x[/tex]
5). [tex]f(x) =(24)^{\frac{1}{4}x}[/tex]
Simplify each of the given exponential function:
1). [tex]f(x) = 5(\sqrt[3]{16})^x = 5(\sqrt[3]{2\times2^3})^x= 5(2\sqrt[3]{2})^x[/tex]
2). [tex]f(x) = 2.3(8)^{\frac{1}{2}x}= 2.3(2^3)^{\frac{1}{2}x}=2.3(2(2)^{\frac{1}{2}})^x[/tex]
3). [tex]f(x) = (81)^{\frac{x}{4}}=(3^4)^{\frac{x}{4}}= (3)^{x}[/tex]
4). [tex]f(x)= \dfrac{3}{4}(\sqrt{27})^x = \dfrac{3}{4}(\sqrt{3^3})^x=\dfrac{3}{4}(3\sqrt{3})^x[/tex]
5). [tex]f(x) =(24)^{\frac{1}{4}x}=(3\times 2^3)^{\frac{1}{4}x}[/tex]
Therefore, the correct options are 1), 2), 3) and 4).
For more information, refer the link given below:
https://brainly.com/question/14355665
