Question

Which exponential functions have been simplified correctly check all that apply

Answer 1

Exponential functions and their inverses like the natural logarithm are inverse functions, meaning they can undo each other. Simplifications like ln(e^x) result in x, and e^(ln x) simplifies to x. This property is integral to correctly simplifying exponential functions.

Step-by-step explanation:

The exponential functions and their inverses, like the natural logarithm (ln), have specific properties where they can "undo" each other. This relationship allows us to simplify expressions involving exponentials and logarithms. For example, when you have ln(ex), this simplifies to x because the natural logarithm and the exponential function are inverse functions.

Similarly, when applying the exponential function to a natural logarithm, like eln(x), the result is just x. Furthermore, when taking square roots of exponentials, you can adjust the exponent to be evenly divisible by 2 and then take the square root of the base number, followed by dividing the exponent by 2. Also, while dividing exponentials, you subtract the exponents when the bases are the same.

Here is an example of this simplification: to compute the square root of e4, we can write it as e4/2, which simplifies to e2.

Answer 2

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).
`f(x)=5(\sqrt[3]{16})^(x)=5(\sqrt[3]{2^(4)})^(x)=5(2\sqrt[3]{2})^(x)`

B).
`f(x)=2.3(8^{(1)/(2)})^(x)=2.3(2.√(2))^(x)\\eq 2.3(4^(x))`

C).
`f(x)=81^{(x)/(4)}=(3^(4))^{(x)/(4)}=(3^{(4)/(4)})^(x)=3^(x)`

D).
`f(x)=(3)/(4)(√(27))^(x)=(3)/(4)(\sqrt{3^(3)})^(x)=(3)/(4).(3√(3))^(x)`

`f(x)=(24)^{(1)/(3)x}=(2^(3).3)^{(1)/(3)x}=(2.3^{(1)/(3)})^(x)\\eq 2(\sqrt[3]{3})^(x)`