Find the units of $\Bbb Z \times \Bbb Z$ and $\Bbb Z \times \Bbb Q$.
Let $R$ and $S$ be rings. Then we always have for the groups of units that $$(R\times S)^\times = R^\times\times S^\times$$ This just follows from the definition of direct products, i.e., of the multiplication of elements of a direct product.