Calculate: (2^{(n^2)})/(2^n * 2^6)=1

Expression: $\frac{ {2}^{\left( {n}^{2} \right)} }{ {2}^{n} \times {2}^{6} }=1$

Determine the defined range

$\begin{array} { l }\frac{ {2}^{\left( {n}^{2} \right)} }{ {2}^{n} \times {2}^{6} }=1,& n \in ℝ\end{array}$

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

$\begin{array} { l }n_1=-2,& n_2=3\end{array}$