Simplify: (u-v)(u+v) + v/u - (3uv-u2)/v2

To simplify the given expression, we will perform the operations step by step.

The expression from the image is: $(u-v)(u+v) + \frac{v}{u} - \frac{3uv - u^2}{v^2}$

Step 1: Expand the first term using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$. $(u-v)(u+v) = u^2 - v^2$ The expression becomes: $u^2 - v^2 + \frac{v}{u} - \frac{3uv - u^2}{v^2}$

Step 2: Find a common denominator for all terms. The denominators are $1$, $u$, and $v^2$. The least common multiple (LCM) of these is $uv^2$. We will rewrite each term with the common denominator $uv^2$.

• For $u^2 - v^2$: $u^2 - v^2 = \frac{(u^2 - v^2) \cdot uv^2}{uv^2} = \frac{u^3v^2 - uv^4}{uv^2}$

• For $\frac{v}{u}$: $\frac{v}{u} = \frac{v \cdot v^2}{u \cdot v^2} = \frac{v^3}{uv^2}$

• For $\frac{3uv - u^2}{v^2}$: $\frac{3uv - u^2}{v^2} = \frac{(3uv - u^2) \cdot u}{v^2 \cdot u} = \frac{3u^2v - u^3}{uv^2}$

Step 3: Combine all terms over the common denominator. $\frac{u^3v^2 - uv^4}{uv^2} + \frac{v^3}{uv^2} - \frac{3u^2v - u^3}{uv^2}$ $\frac{u^3v^2 - uv^4 + v^3 - (3u^2v - u^3)}{uv^2}$

Step 4: Distribute the negative sign in the numerator and simplify. $\frac{u^3v^2 - uv^4 + v^3 - 3u^2v + u^3}{uv^2}$ There are no like terms to combine in the numerator.

The simplified expression is: \frac{u^3v^2 - uv^4 + v^3 - 3u^2v + u^3{uv^2}}