Answer
The solution set is $\left\{-1, 7\right\}$.
Work Step by Step
To solve the given equation, make the two sides have the same base.
Note that $27=3^3$, so the given equation is equivalent to:
$3^{x^2-7}=(3^3)^{2x}$
Use the rule $(a^m)^n = a^{mn}$ to obtain:
$3^{x^2-7} = 3^{3(2x)}
\\3^{x^2-7}=3^{6x}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$x^2-7=6x$
Subtract $6x$ to both sides of the equation to obtain:
$\begin{array}{ccc}
&x^2-7-6x &= &6x-6x
\\&x^2-6x-7 &= &0
\end{array}$
Factor the trinomial to obtain:
$(x-7)(x+1)=0$
Equate each factor to zero, and then solve each equation to obtain:
$\begin{array}{ccc}
&x-7=0 &\text{ or } &x+1-0
\\&x=7 &\text{ or } &x=-1
\end{array}$
Thus, the solution set is $\left\{-1, 7\right\}$.
