$\begin{array} { l }\frac{ {2}^{\left( {n}^{2} \right)} }{ {2}^{n} \times {2}^{6} }=1,& n \in ℝ\end{array}$
Multiply the terms with the same base by adding their exponents$\frac{ {2}^{\left( {n}^{2} \right)} }{ {2}^{n+6} }=1$
Simplify the expression${2}^{{n}^{2}-n-6}=1$
Write the number in exponential form with the base of $2$${2}^{{n}^{2}-n-6}={2}^{0}$
Since the bases are the same, set the exponents equal${n}^{2}-n-6=0$
Write $-n$ as a difference${n}^{2}+2n-3n-6=0$
Factor out $n$ from the expression$n \times \left( n+2 \right)-3n-6=0$
Factor out $-3$ from the expression$n \times \left( n+2 \right)-3\left( n+2 \right)=0$
Factor out $n+2$ from the expression$\left( n+2 \right) \times \left( n-3 \right)=0$
When the product of factors equals $0$, at least one factor is $0$$\begin{array} { l }n+2=0,\\n-3=0\end{array}$
Solve the equation for $n$$\begin{array} { l }n=-2,\\n-3=0\end{array}$
Solve the equation for $n$$\begin{array} { l }n=-2,\\n=3\end{array}$
The equation has $2$ solutions$\begin{array} { l }n_1=-2,& n_2=3\end{array}$