$$ \begin{array}{cll} \displaystyle \frac{ 6 r - 6 }{ 4 r - 5}-\frac{ 3 r - 5 }{ 4 r ^2 + 3 r - 10 }\div \frac{ 5 - 3 r } {r + 2 } &= \displaystyle \frac{ 6 r - 6 }{ 4 r - 5}-\frac{ 3 r - 5 }{ 4 r ^2 + 3 r - 10 }\cdot \frac{r + 2 }{ 5 - 3 r } &\mbox{invert and multiply}\\ &= \displaystyle \frac{ 6 r - 6 }{ 4 r - 5}-\frac{ \color{magenta}{(3 r - 5)}\cdot 1 }{ (4r-5)\color{magenta}{(r+2)} }\cdot \frac{\color{magenta}{(r + 2)}\cdot 1 }{-1\cdot\color{magenta}{(3r-5)}} &\mbox{factor like crazy}\\ &= \displaystyle \frac{ 6 r - 6 }{ 4 r - 5}-\frac{ 1 }{ 4r-5 }\cdot \frac{ 1 }{-1} &\mbox{cancel common factors}\\ &= \displaystyle \frac{ 6 r - 6 }{ 4 r - 5}+\frac{ 1 }{ 4r-5 } &\mbox{simplify}\\ &= \displaystyle \frac{ 6 r - 6 + 1 }{ 4r-5 } &\mbox{add over common denom}\\ &= \displaystyle \frac{ 6 r - 5 }{ 4r-5 } &\mbox{simplify}\\ \end{array} $$

$$ \begin{array}{cll} \displaystyle \frac{ \displaystyle \frac{k^2-8k+15}{k^2-10k+25} }{ \displaystyle \frac{k^2-6k+9}{11 k^2-275} } &= \displaystyle \displaystyle \frac{k^2-8k+15}{k^2-10k+25} \div \displaystyle \frac{k^2-6k+9}{11 k^2-275} &\mbox{turn into division problem}\\ &= \displaystyle \frac{k^2-8k+15}{k^2-10k+25} \cdot \displaystyle \frac{11 k^2-275}{k^2-6k+9} &\mbox{invert and multiply}\\ &= \displaystyle \frac{(k-3)(k-5)}{(k-5)(k-5)} \cdot \displaystyle \frac{11(k^2-25)}{(k-3)(k-3)} &\mbox{factor like crazy}\\ &= \displaystyle \frac{\color{magenta}{(k-3)(k-5)}}{\color{magenta}{(k-5)(k-5)}} \cdot \displaystyle \frac{11(k+5)\color{magenta}{(k-5)}}{\color{magenta}{(k-3)}(k-3)} &\mbox{identify } \color{magenta}{\mbox{common}} \mbox{ factors}\\ &= \displaystyle \displaystyle \frac{11(k+5)}{k-3} &\mbox{cancel } \color{magenta}{\mbox{common}} \mbox{ factors}\\ \end{array} $$

We did this one already!

We did this one already!

$$ \begin{array}{cll} \displaystyle \frac{ \displaystyle \frac{1}{n t^2}+\displaystyle \frac{1}{n^2t} }{ \displaystyle \frac{1}{t}+\displaystyle \frac{1}{n} } &= \displaystyle \frac{ \displaystyle \frac{1}{n t^2}+\displaystyle \frac{1}{n^2t} }{ \displaystyle \frac{1}{t}+\displaystyle \frac{1}{n} }\cdot \color{magenta}{1} &\mbox{multiply by 1}\\ &= \displaystyle \frac{ \displaystyle \frac{1}{n t^2}+\displaystyle \frac{1}{n^2t} }{ \displaystyle \frac{1}{t}+\displaystyle \frac{1}{n} }\cdot \color{magenta}{\frac{n^2t^2}{n^2t^2}} &\mbox{multiply by fancy 1}\\ &= \displaystyle \frac{ \displaystyle \left(\frac{1}{n t^2}+\displaystyle \frac{1}{n^2t}\right) \cdot \color{magenta}{n^2t^2} }{ \displaystyle \left(\frac{1}{t}+\displaystyle \frac{1}{n}\right)\cdot \color{magenta}{n^2t^2} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle \frac{1}{n t^2}\cdot \color{magenta}{n^2t^2}+\displaystyle \frac{1}{n^2t}\cdot \color{magenta}{n^2t^2} }{ \displaystyle \frac{1}{t}\cdot \color{magenta}{n^2t^2}+\displaystyle \frac{1}{n}\cdot \color{magenta}{n^2t^2} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle n+t }{ \displaystyle n^2t+nt^2 } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle n+t }{ \displaystyle nt(n+t) } &\mbox{factor out GCF in denom}\\ &= \displaystyle \frac{ \displaystyle 1\cdot \color{magenta}{(n+t)}}{ \displaystyle nt\color{magenta}{(n+t)} } &\mbox{identify } \color{magenta}{\mbox{common}} \mbox{ factors}\\ &= \displaystyle \frac{ \displaystyle 1 }{ \displaystyle nt } &\mbox{cancel } \color{magenta}{\mbox{common}} \mbox{ factors}\\ \end{array} $$

$$ \begin{array}{cll} \displaystyle \frac{ v-\displaystyle \frac{1}{x} }{ x-\displaystyle \frac{1}{v} } &= \displaystyle \frac{ v-\displaystyle \frac{1}{x} }{ x-\displaystyle \frac{1}{v} }\cdot \color{magenta}{1} &\mbox{multiply by 1}\\ &= \displaystyle \frac{ v-\displaystyle \frac{1}{x} }{ x-\displaystyle \frac{1}{v} }\cdot \color{magenta}{\frac{xv}{xv}} &\mbox{multiply by fancy 1}\\ &= \displaystyle \frac{ \displaystyle \left(v-\displaystyle \frac{1}{x}\right) \cdot \color{magenta}{xv} }{ \displaystyle \left(x-\displaystyle \frac{1}{v}\right)\cdot \color{magenta}{xv} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle v\cdot \color{magenta}{xv}-\displaystyle \frac{1}{x}\cdot \color{magenta}{xv} }{ \displaystyle x\cdot \color{magenta}{xv}-\displaystyle \frac{1}{v}\cdot \color{magenta}{xv} } &\mbox{}\\ &= \displaystyle \frac{ xv^2-v}{ x^2v-x } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle v(xv-1) }{x(xv-1)} &\mbox{factor out GCFs}\\ &= \displaystyle \frac{ \displaystyle v\color{magenta}{(xv-1)} }{x\color{magenta}{(xv-1)}} &\mbox{identify } \color{magenta}{\mbox{common}} \mbox{ factors}\\ &= \displaystyle \frac{ \displaystyle v }{ \displaystyle x } &\mbox{cancel } \color{magenta}{\mbox{common}} \mbox{ factors}\\ \end{array} $$

$$ \begin{array}{cll} \displaystyle \frac{ \displaystyle \frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} }{ \displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} } &= \displaystyle \frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} \div \displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} &\mbox{re-express as division problem}\\ &= \displaystyle \left(\frac{ \displaystyle 1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} \cdot \color{magenta}{1}\right) \div \left(\displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} \cdot \color{magenta}{1}\right) &\mbox{multiply by 1}\\ &= \displaystyle \left(\frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} \cdot \color{magenta}{\frac{x}{x}}\right) \div \left(\displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} \cdot \color{magenta}{\frac{p}{p}}\right) &\mbox{turn into fancy 1s}\\ &= \displaystyle \frac{\left(1+\displaystyle \frac{p}{x}\right)\cdot \color{magenta}{x}}{\left(1-\displaystyle \frac{p}{x}\right)\cdot \color{magenta}{x}} \div \displaystyle \frac{(p+x)\cdot \color{magenta}{p}}{\left(p-\displaystyle \frac{x^2}{p}\right)\cdot \color{magenta}{p}} &\mbox{}\\ &= \displaystyle \frac{1\cdot \color{magenta}{x}+\displaystyle \frac{p}{x}\cdot \color{magenta}{x}}{1\cdot \color{magenta}{x}-\displaystyle \frac{p}{x}\cdot \color{magenta}{x}} \div \displaystyle \frac{p\cdot \color{magenta}{p}+x\cdot \color{magenta}{p}}{p\cdot \color{magenta}{p}-\displaystyle \frac{x^2}{p}\cdot \color{magenta}{p}} &\mbox{}\\ &= \displaystyle \frac{1\cdot \color{magenta}{x}+\displaystyle \frac{p}{x}\cdot \color{magenta}{x}}{1\cdot \color{magenta}{x}-\displaystyle \frac{p}{x}\cdot \color{magenta}{x}} \div \displaystyle \frac{p\cdot \color{magenta}{p}+x\cdot \color{magenta}{p}}{p\cdot \color{magenta}{p}-\displaystyle \frac{x^2}{p}\cdot \color{magenta}{p}} &\mbox{}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \div \displaystyle \frac{p^2+xp}{p^2-x^2} &\mbox{}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \div \displaystyle \frac{p\color{magenta}{(p+x)}}{\color{magenta}{(p+x)}(p-x)} &\mbox{factor}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \div \displaystyle \frac{p}{p-x} &\mbox{cancel common factors}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \cdot \displaystyle \frac{p-x}{p} &\mbox{invert and multiply}\\ &= \displaystyle \displaystyle \frac{x+p}{\color{magenta}{x-p}}\cdot \displaystyle \frac{-(\color{magenta}{x-p})}{p} &\mbox{factor out negative}\\ &= \displaystyle \displaystyle \frac{x+p}{1}\cdot \displaystyle \frac{-1}{p} &\mbox{cross cancel}\\ &= \displaystyle -\frac{x+p}{p} &\mbox{simplify}\\ \end{array} $$ Alternative: You may also leave everything in complex-fraction form and re-express as a division problem near the end of the calculation. $$ \begin{array}{cll} \displaystyle \frac{ \displaystyle \frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} }{ \displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} } &= \displaystyle \frac{ \displaystyle \frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} \cdot \color{magenta}{1} }{ \displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} \cdot \color{magenta}{1} } &\mbox{multiply by 1}\\ &= \displaystyle \frac{ \displaystyle \frac{1+\displaystyle \frac{p}{x}}{1-\displaystyle \frac{p}{x}} \cdot \color{magenta}{\frac{x}{x}} }{ \displaystyle \frac{p+x}{p-\displaystyle \frac{x^2}{p}} \cdot \color{magenta}{\frac{p}{p}} } &\mbox{turn into fancy 1s}\\ &= \displaystyle \frac{ \displaystyle \frac{\left(1+\displaystyle \frac{p}{x}\right)\cdot \color{magenta}{x}}{\left(1-\displaystyle \frac{p}{x}\right)\cdot \color{magenta}{x}} }{ \displaystyle \frac{(p+x)\cdot \color{magenta}{p}}{\left(p-\displaystyle \frac{x^2}{p}\right)\cdot \color{magenta}{p}} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle \frac{1\cdot \color{magenta}{x}+\displaystyle \frac{p}{x}\cdot \color{magenta}{x}}{1\cdot \color{magenta}{x}-\displaystyle \frac{p}{x}\cdot \color{magenta}{x}} }{ \displaystyle \frac{p\cdot \color{magenta}{p}+x\cdot \color{magenta}{p}}{p\cdot \color{magenta}{p}-\displaystyle \frac{x^2}{p}\cdot \color{magenta}{p}} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle \frac{1\cdot \color{magenta}{x}+\displaystyle \frac{p}{x}\cdot \color{magenta}{x}}{1\cdot \color{magenta}{x}-\displaystyle \frac{p}{x}\cdot \color{magenta}{x}} }{ \displaystyle \frac{p\cdot \color{magenta}{p}+x\cdot \color{magenta}{p}}{p\cdot \color{magenta}{p}-\displaystyle \frac{x^2}{p}\cdot \color{magenta}{p}} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle \frac{x+p}{x-p} }{ \displaystyle \frac{p^2+xp}{p^2-x^2} } &\mbox{}\\ &= \displaystyle \frac{ \displaystyle \frac{x+p}{x-p} }{ \displaystyle \frac{p\color{magenta}{(p+x)}}{\color{magenta}{(p+x)}(p-x)} } &\mbox{factor}\\ &= \displaystyle \frac{ \displaystyle \frac{x+p}{x-p} }{ \displaystyle \frac{p}{p-x} } &\mbox{cancel common factors}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \div \displaystyle \frac{p}{p-x} &\mbox{re-express as division problem}\\ &= \displaystyle \displaystyle \frac{x+p}{x-p} \cdot \displaystyle \frac{p-x}{p} &\mbox{invert and multiply}\\ &= \displaystyle \displaystyle \frac{x+p}{\color{magenta}{x-p}} \cdot \displaystyle \frac{-(\color{magenta}{x-p})}{p} &\mbox{factor out negative}\\ &= \displaystyle \displaystyle \frac{x+p}{1} \cdot \displaystyle \frac{-1}{p} &\mbox{cross cancel}\\ &= \displaystyle -\frac{x+p}{p} &\mbox{simplify}\\ \end{array} $$

$$ \begin{array}{cll} \displaystyle R &= \displaystyle \frac{1}{\displaystyle \frac{1}{r_1}+\displaystyle \frac{1}{r_2}} &\mbox{state general formula}\\ &= \displaystyle \frac{1}{\displaystyle \frac{1}{41}+\displaystyle \frac{1}{8}} &\mbox{substitute in known values}\\ &= \displaystyle \frac{1}{\displaystyle \frac{1}{41}\cdot\color{magenta}{\frac{8}{8}}+\displaystyle \frac{1}{8}\cdot\color{magenta}{\frac{41}{41}}} &\mbox{multiply by fancy ones}\\ &= \displaystyle \frac{1}{\displaystyle \frac{8}{328}+\displaystyle \frac{41}{328}} &\mbox{get common denom}\\ &= \displaystyle \frac{1}{\displaystyle \frac{8+41}{328}} &\mbox{add over common denom}\\ &= \displaystyle \frac{1}{\displaystyle \frac{49}{328}} &\mbox{simplify}\\ &= \displaystyle 1\div \frac{49}{328} &\mbox{re-express as division problem}\\ &= \displaystyle 1 \cdot \frac{328}{49} &\mbox{invert and multiply}\\ &= \displaystyle \frac{328}{49} &\mbox{}\\ &\approx\displaystyle 6.69 &\mbox{approximate as a decimal}\\ \end{array} $$ The total resistance of the circuit is about $6.69$ ohms.