$bu_*(B\Z/p)^n$ as a Graded Group

Abstract

Let $p$ be a prime. We calculate $bu_*(B\Z/p)^n$, the connective unitary $K-$theory of the $n-$fold smash product of the classifying space for the cyclic group of order $p$, as a graded group using a K\"{u}nneth formula short exact sequence for $n=2$ and inductively for any $n\geq 2$.
While this smashing is in progress some other spectra appear, for instance, the spectrum $H\Z/p\wedge (B\Z/p)^r$ for $r<n$. In order to producing a new homotopy equivalent to $bu\wedge (B\Z/p)^n$, we need to find a homotopy equivalence which simplifies the spectrum $H\Z/p\wedge (B\Z/p)^r$.