Solve: 1/(mn) - (mn-(m:1/n+n/m:(mn))) Complex Fraction Problem

To solve this problem, follow these steps:

• Simplify the innermost portion of the expression: $(m:\frac{1}{n}+\frac{n}{m}:(mn))$.
• We interpret $m:\frac{1}{n}$ as $m \cdot n$, translating division by a fraction into multiplication.
• The entire sub-expression becomes $(m \cdot n + \frac{n}{m} \cdot \frac{1}{mn})$.
• Recognize that $\frac{n}{m} \cdot \frac{1}{mn} = \frac{n}{m^2n} = \frac{1}{m^2}$.
• This simplifies to $mn + \frac{1}{m^2}$.

Now simplify the outer expression:

• Substitute back into the original expression:
• The expression becomes $\frac{1}{mn} - (mn - (mn + \frac{1}{m^2}))$.
• Calculate inside the bracket: $mn - mn - \frac{1}{m^2} = -\frac{1}{m^2}$.
• Therefore, the expression becomes $\frac{1}{mn} - (-\frac{1}{m^2}) = \frac{1}{mn} + \frac{1}{m^2}$.

Therefore, the simplified form of the problem is $\frac{1}{mn} + \frac{1}{m^2}$.