  We employ a generalised version of Heath-Brown's square sieve in order to
establish an asymptotic estimate of the number of solutions $a, b \in \mathbb
N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is
$l$-free. This is the first time that this problem has been studied with
distinct powers $k$ and $l$.
