Answer:
The base is: ![3 \sqrt[3]{4}](https://tex.z-dn.net/?f=3%20%5Csqrt%5B3%5D%7B4%7D)
Step-by-step explanation:
Given
![f(x) = \frac{1}{4}(\sqrt[3]{108})^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%28%5Csqrt%5B3%5D%7B108%7D%29%5Ex)
Required
The base
Expand 108
![f(x) = \frac{1}{4}(\sqrt[3]{3^3 * 4})^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%28%5Csqrt%5B3%5D%7B3%5E3%20%2A%204%7D%29%5Ex)
Rewrite the exponent as:

Expand


Rewrite as:
![f(x) = \frac{1}{4}(3 \sqrt[3]{4})^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%283%20%5Csqrt%5B3%5D%7B4%7D%29%5Ex)
An exponential function has the following form:

Where

By comparison:
![b =3 \sqrt[3]{4}](https://tex.z-dn.net/?f=b%20%3D3%20%5Csqrt%5B3%5D%7B4%7D)
So, the base is: ![3 \sqrt[3]{4}](https://tex.z-dn.net/?f=3%20%5Csqrt%5B3%5D%7B4%7D)
The rate of the passenger train is 81.67 MPH and the rate of the freight train is 66.67 MPH.
<h3>What is a linear equation?</h3>
It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
We have:
The distance traveled by passenger train = 245 miles
The distance traveled by freight train = 245 miles
The time takes for both trains is the same.
Let's suppose the rate of the passenger train is x and the rate of the freight train is y;
As we know,
Distance = speed×time
245/x = 200/y
200x = 245y
200x - 245y = 0 ..(1)
x = 15 + y ..(2)
Equations (1) and (2) represent the situation.
After solving equations (1) and (2):

Thus, the rate of the passenger train is 81.67 MPH and the rate of the freight train is 66.67 MPH.
Learn more about the linear equation here:
brainly.com/question/11897796
#SPJ1
Answer is A.
P−⁸ × Q −²⁰ × P⁸ × Q−¹⁰
Q−³⁰
1/q³⁰
Both expressions are related because 12 divided by 4 equals to 3, and 1/4 of 12 equals to 3 too.
Step-by-step explanation:
kindly see attached picture
hope it helped you:)