The minimum value of $2^{{\left(x^2-3\right)}^3+27}$ is

Function $2^{{\left(x^2-3\right)}^3}+27$ is minimum when ${\left(x^2-3\right)}^3+27$ is minimum

Clearly $27+{\left(x^2-3\right)}^3=x^2\left\{{\left(x^2-\frac{9}{2}\right)}^2+\frac{27}{4}\right\}\ge 0$ for all x. 

Therefore, the minimum value of  ${\left(x^2-3\right)}^3+27$ is zero for .$x = 0$ Hence, the minimum value of the given function is $2° = 1.$