Let G be a group and γ ∈ G'. The minimal number of squares which is required to write γ as a product of squares in G is called the square length of γ denoted by Sq(γ). We consider certain equations over free group F_2 = F (x_1 , x_2). Using this, we find Sq([x_2~r,x_1~s]~k) where r, s, k ∈ Z. We also examine the problem in free nilpotent groups 〈x_1, x_2〉. In this case, we find explicit formula to write [x_2~r,x_1~8]~ k as a product of minimal number of squares. We prove that in a free nilpotent group it is possible to write certain nontrivial commutators as a proper power. We also describe solutions of the following equations in a free nilpotent group:[x_2~r,x_1~s]~k= u ~ 2 , [ x_2~r , x_1~s ]~k = u_1~2 u_2~2 , [x_ 2~r, x_1~s ]~k=u_1~2u_2~2u_3~2 in which k is an odd integer and r, s, and k ∈Z.
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