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Let 
$w_{1}$ and 
$w_{2}$ be nontrivial words in free groups 
$F_{n_{1}}$ and 
$F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups 
$G$, there exist subsets 
$C_{1}\subseteq w_{1}(G)$ and 
$C_{2}\subseteq w_{2}(G)$ such that 
$|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and 
$C_{1}C_{2}=G$. In particular, if 
$w$ is any nontrivial word and 
$G$ is a sufficiently large finite nonabelian simple group, then 
$w(G)$ contains a thin base of order 
$2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of 
$G$ of order 
$2$ are established for any finite group and for any compact Lie group 
$G$.
