  Let $p$ be a prime number, $m$ be an even positive integer, and
$\mathbb{F}_q$ be a finite field with $q = p^m$ elements. In this paper, we
compute the number of solutions with all coordinates in $\mathbb{F}_q^*$ of
equations of the form $$a_1 x_1^{d} + \dots + a_s x_s^{d} = b, \quad a_i \in
\mathbb{F}_q^*, \, b \in \mathbb{F}_q,$$ when the coefficients and exponents
satisfy arithmetic conditions that allow the computation through pure Gauss
sums. We then apply this result to compute the number of solutions of equations
of the form \begin{equation*}
  a_1 x_1^{d_{1,1}} \cdots x_n^{d_{n,1}} + \dots + a_s x_1^{d_{1,s}}\cdots
x_n^{d_{n,s}} = b, \end{equation*} where all exponents are positive, and the
equation is related in a certain way to a diagonal equation with the
characteristics we have mentioned.
