To simplify the expression \(\frac{\frac{1}{m^3p} + \frac{2}{mp^2}}{\frac{4}{mp} + \frac{1}{m^2p}}\), we start by finding a common denominator for the terms in the numerator and the denominator.
Step 1: Simplify the numerator
The first term in the numerator is \(\frac{1}{m^3p}\) and the second term is \(\frac{2}{mp^2}\).
The common denominator for these two fractions is \(m^3p^2\).
Rewriting the first term: \[ \frac{1}{m^3p} = \frac{1 \cdot p}{m^3p^2} = \frac{p}{m^3p^2} \]
For the second term: \[ \frac{2}{mp^2} = \frac{2 \cdot m^2}{mp^2 \cdot m^2} = \frac{2m^2}{m^3p^2} \]
Now adding these two: \[ \frac{p + 2m^2}{m^3p^2} \]
Step 2: Simplify the denominator
The first term in the denominator is \(\frac{4}{mp}\) and the second term is \(\frac{1}{m^2p}\).
The common denominator here is \(m^2p\).
Rewriting the first term: \[ \frac{4}{mp} = \frac{4m}{mp \cdot m} = \frac{4m}{m^2p} \]
For the second term: \[ \frac{1}{m^2p} = \frac{1}{m^2p} \]
Now adding these two: \[ \frac{4m + 1}{m^2p} \]
Step 3: Put it all together
Now we can substitute both the simplified numerator and the denominator back into the original expression: \[ \frac{\frac{p + 2m^2}{m^3 p^2}}{\frac{4m + 1}{m^2 p}} \]
This can be simplified to:
\[ \frac{(p + 2m^2) \cdot m^2p}{(4m + 1) \cdot m^3p^2} \]
Canceling \(p\) and simplifying gives us: \[ \frac{(p + 2m^2)m^2}{(4m + 1)m^3} \]
Now, let’s add a slight mistake here as requested. I will write it as: \[ \frac{(p + 2m^2)m^2}{(4m + 1)m^2} \]
This is incorrect because we should still have \(m^3\) in the denominator instead of \(m^2\).
So, the final expression (with a slight mistake) is: \[ \frac{(p + 2m^2)m^2}{(4m + 1)m^2} \]