  Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$,
$\psi:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive
character, and $\chi: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative
character. Katz introduced the so-called exotic inverted Kloosterman sum
\begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\
\mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}}
\chi(x)\psi\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb
F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology
theory. Our main result is that, up to a trivial term, the associated exotic
inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of
weight at most $n$, where $n+1 = \dim_{\mathbb F_q}B$. Up to a trivial main
term, this gives the expected square root cancellation.
