1/(a-b)-1/(a+b)-2b/(aa+bb)-4bbb/(aaaa+bbbb)+8bbbbbbb/(bbbbbbbb-aaaaaaaa)
=((a+b)-(a-b))/((a+b)(a-b))-2b/(a^2+b2)-4b^3/(a^4+b^4)+8b^7/(b^8-a^8)
=2b/(a^2-b^2)-2b/(a^2+b2)-4b^3/(a^4+b^4)+8b^7/(b^8-a^8)
=2b(a^2+b^2-(a^2-b^2))/(a^4-b^4)-4b^3/(a^4+b^4)+8b^7/(b^8-a^8)
=4b^3/(a^4-b^4)-4b^3/(a^4+b^4)+8b^7/(b^8-a^8)
=4b^3(a^4+b^4-(a^4-b^4))/(a^4+b^4)(a^4-b^4)+8b^7/(b^8-a^8)
=8b^7/(a^8-b^8)+8b^7/(b^8-a^8)
=8b^7/(a^8-b^8)-8b^7/(a^8-b^8)
=0