Solve: mn-(3mn-n/m:m/n+nm/3:m²) Complex Algebraic Expression

To solve this expression, we will simplify it step by step:

The expression to simplify is:

$mn - \left(3mn - \frac{n}{m} : \frac{m}{n} + \frac{nm}{3} : m^2 \right)$

First, simplify each term within the parentheses:

• The division $\frac{n}{m} : \frac{m}{n}$ can be rewritten as $\frac{n}{m} \cdot \frac{n}{m}$ because $\frac{m}{n}$ inverted becomes $\frac{n}{m}$. This results in $\frac{n^2}{m^2}$.
• Next, the operation $\frac{nm}{3} : m^2$ is rewritten as $\frac{nm}{3} \cdot \frac{1}{m^2}$. This simplifies to $\frac{n}{3m}$ because $m \cdot m = m^2$.

Substitute these simplified forms back into the expression inside the parentheses:

$3mn - \frac{n^2}{m^2} + \frac{n}{3m}$

Now rewrite the entire expression outside of the parentheses and solve:

$mn - \left( 3mn - \frac{n^2}{m^2} + \frac{n}{3m} \right)$

Use distributive law to expand:

$mn - 3mn + \frac{n^2}{m^2} - \frac{n}{3m}$

Combine like terms:

$-2mn + \frac{n^2}{m^2} - \frac{n}{3m}$

Therefore, the solution to the simplified expression is:

$-2mn + \frac{n^2}{m^2} - \frac{n}{3m}$