 doturitip9

2022-07-10

How to simplify an expression?
I have tried to simplify this expression for quite a long time now but I can't find how to do it.
$\left(\frac{1}{2+4m}-\frac{1-m}{8{m}^{3}+1}:\frac{1-2m}{2{m}^{2}-2m+1}\right)\cdot \frac{4m+2}{2m-1}-\frac{1}{1-4m+4{m}^{2}}$
Can someone help me with it? Shawn Castaneda

Expert

Starting with the original expression:
$\left(\frac{1}{2+4m}-\frac{1-m}{8{m}^{3}+1}:\frac{1-2m}{2{m}^{2}-2m+1}\right)\cdot \frac{4m+2}{2m-1}-\frac{1}{1-4m+4{m}^{2}}$
Factoring where possible and changing the order of some terms:
$\left(\frac{1}{2\left(2m+1\right)}+\frac{m-1}{\left(2m+1\right)\left(4{m}^{2}-2m+1\right)}:\frac{-\left(2m-1\right)}{2{m}^{2}-2m+1}\right)\cdot \frac{2\left(2m+1\right)}{2m-1}-\frac{1}{\left(2m-1{\right)}^{2}}$
Converting division into multiplication by the reciprocal; distributing the outside fraction:
$\left(\frac{1}{2\left(2m+1\right)}\frac{2\left(2m+1\right)}{2m-1}+\frac{m-1}{\left(2m+1\right)\left(4{m}^{2}-2m+1\right)}\frac{2{m}^{2}-2m+1}{-\left(2m-1\right)}\frac{2\left(2m+1\right)}{2m-1}\right)-\frac{1}{\left(2m-1{\right)}^{2}}$
Some cancellations:
$\phantom{\rule{-0.8mm}{0ex}}\left(\frac{1}{\overline{)2\left(2m+1\right)}}\frac{\overline{)2\left(2m+1\right)}}{2m-1}+\frac{m-1}{\overline{)\left(2m+1\right)}\left(4{m}^{2}-2m+1\right)}\frac{2{m}^{2}-2m+1}{-\left(2m-1\right)}\frac{2\overline{)\left(2m+1\right)}}{2m-1}\right)-\frac{1}{\left(2m-1{\right)}^{2}}$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\frac{1}{2m-1}-\frac{2\left(m-1\right)}{4{m}^{2}-2m+1}\frac{2{m}^{2}-2m+1}{\left(2m-1{\right)}^{2}}-\frac{1}{\left(2m-1{\right)}^{2}}$
Finding common denominators:
$\frac{8{m}^{3}-8{m}^{2}+4m-1}{\left(4{m}^{2}-2m+1\right)\left(2m-1{\right)}^{2}}-\frac{4{m}^{3}-8{m}^{2}+6m-2}{\left(4{m}^{2}-2m+1\right)\left(2m-1{\right)}^{2}}-\frac{4{m}^{2}-2m+1}{\left(4{m}^{2}-2m+1\right)\left(2m-1{\right)}^{2}}$
$\frac{4{m}^{3}-4{m}^{2}}{\left(4{m}^{2}-2m+1\right)\left(2m-1{\right)}^{2}}$
Factoring one last time:
$\frac{4{m}^{2}\left(m-1\right)}{\left(4{m}^{2}-2m+1\right)\left(2m-1{\right)}^{2}}$

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