MHB How Do You Simplify Complex Fractional Expressions?

[karush]

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

$$=\frac{u^3 v+4{u}^{3}-{u}^{2}v^3}
{{u}^{4 }v+4{u}^{4}+v^4+4{v}^{2 } }
$$
Steps: Common denomator, Mutiply by reciprocal, Factor out v

I have done this 5 times and get different answers

 

[MarkFL]

Okay, we begin with:

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

We see the LCD is:

$$u^2v(v+4)$$

And so we write:

$$\frac{\dfrac{u}{v}-\dfrac{{v}^{2 }}{v+4}}{\dfrac{{u}^{2}}{v}+\dfrac{v}{{u}^{2}}}\cdot\frac{u^2v(v+4)}{u^2v(v+4)}$$

Distributing, we obtain:

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

Distribute:

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

 

[karush]

Well that's a much better way