  In this paper, we study the Erd\H{o}s-Falconer distance problem in five
dimensions for sets of Cartesian product structure. More precisely, we show
that for $A\subset \mathbb{F}_p$ with $|A|\gg p^{\frac{13}{22}}$, then
$\Delta(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we obtain stronger statements
as follows:
  If $|A|\gg p^{\frac{13}{22}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$
  If $|A|\gg p^{\frac{4}{7}}$, then
$(A-A)^2+(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$
