(a -1 +b -1 )(a -1 -b -1 ) ​3

 

 

\((a^{-1}+b^{-1})(a^{-1}-b^{-1})^3\\ =\left(\frac{1}{a}+\frac{1}{b}\right)\times\left(\frac{1}{a}-\frac{1}{b}\right)^3\\ =\left(\frac{b}{ab}+\frac{a}{ab}\right)\times\left(\frac{b}{ab}-\frac{a}{ab}\right)^3\\ =\left(\frac{b+a}{ab}\right)\times\left(\frac{b-a}{ab}\right)^3\\ =\frac{(b+a)(b-a)^3}{a^4b^4}\\ \)

 

You may finish here but you may want to expand it.  

 

\(=\frac{(b+a)(b-a)\times(b-a)^2}{a^4b^4}\\ =\frac{(b^2-a^2)\times(b^2-2ab+a^2)}{a^4b^4}\\ =\frac{b^4-2ab^3+2a^3b-a^4}{a^4b^4}\\ \)

simplify:(a-^1+b^-1)(a^-1-b^-1)^3

 

-((a-b)^3 (a+b)^3)/(a^6 b^6), or

-(a^2-b^2)^3/(a^6 b^6), or

1/a^6-3/(a^4 b^2)+3/(a^2 b^4)-1/b^6