If a/ (a^2+1) =1/3, then determine a^3 / (a^6+a^5+a^4+a^3+a^2+a+1).
This is easier to do if we turn things upside down. Which gives,
If (a^2+1)/a =3, then determine (a^6+a^5+a^4+a^3+a^2+a+1) / a^3.
Developing the expression (a^6+a^5+a^4+a^3+a^2+a+1) / a^3, we get
([a^6 + 3a^4 + 3a^2 + 1] + a^5 - 2a^4 + a^3 - 2a^2 + a) / a^3.
([a^2 + 1]^3 + a^5 - 2a^4 + a^3 - 2a^2 + a) / a^3.
[(a^2 + 1)/a]^3 + a(a^4 - 2a^3 + a^2 - 2a + 1) / a^3.
[(a^2 + 1)/a]^3 + (a^4 - 2a^3 + a^2 - 2a + 1) / a^2
[(a^2 + 1)/a]^3 + ([a^4 + 2a^2 + 1] - 2a^3 - a^2 - 2a) / a^2.
[(a^2 + 1)/a]^3 + [(a^2 + 1)^2] / a^2 - (2a^3 + a^2 + 2a) / a^2.
[(a^2 + 1)/a]^3 + [(a^2 + 1)/a]^2 - (2a^2 + a + 2) / a.
[(a^2 + 1)/a]^3 + [(a^2 + 1)/a]^2 - ([2a^2 + 2] + a) / a.
[(a^2 + 1)/a]^3 + [(a^2 + 1)/a]^2 - 2(a^2 + 1]/a - 1.
Substituting for (a^2 + 1)/a = 3 into the above,
3^3 + 3^2 – 2*3 – 1.
27 + 9 – 6 – 1.
29.
Therefore the original expression, a^3 / (a^6+a^5+a^4+a^3+a^2+a+1), = 1/29