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

$$(u-v)(u+v) = u^2 - v^2$$

The expression becomes:

$$u^2 - v^2 + \frac{v}{u} - \frac{3uv - u^2}{v^2}$$

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 $\frac{3uv - u^2}{v^2}$:

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

$$\frac{u^3v^2 - uv^4}{uv^2} + \frac{v^3}{uv^2} - \frac{3u^2v - u^3}{uv^2}$$ $$\frac{u^3v^2 - uv^4 + v^3 - (3u^2v - u^3)}{uv^2}$$

$$\frac{u^3v^2 - uv^4 + v^3 - 3u^2v + u^3}{uv^2}$$

There are no like terms to combine in the numerator.