Question

What is the product?

StartFraction 3 k Over k + 1 EndFraction times StartFraction k squared minus 1 Over 3 k cubed EndFraction
StartFraction k + 1 Over k squared EndFraction
StartFraction k minus 1 Over k squared EndFraction
StartFraction negative 1 Over k EndFraction
StartFraction 1 Over k EndFraction

Answer 1

Compute
(3k/(k+1)) * ((k^2-1)/(3k^3)) * ((k+1)/k^2) * ((k-1)/k^2) * (−1/k) * (1/k).

Note k^2−1=(k−1)(k+1). Multiplying and canceling gives
numerator: 3k*(k−1)^2*(k+1)^2*(−1),
denominator: 3(k+1)k^9.

Cancel 3, one (k+1), and one k to get

−(k−1)^2(k+1)/k^8.

(Require k ≠ 0, −1.)