General term in the binomial expansion of ${\left(x^{\frac{2}{3}}+\frac{\alpha }{x^3}\right)}^{22}$ is

$T_{r+1}={}^{22}C_r{\left(x^{\frac{2}{3}}\right)}^{22-r}{\left(\frac{\alpha }{x^3}\right)}^r$

$\Rightarrow T_{r+1}={}^{22}C_r{\left(x\right)}^{\frac{44-11r}{3}}{\alpha }^r$

For term independent of $x$, 

$\frac{44-11r}{3}=0$

$\Rightarrow r=4$

So,

$T_5={}^{22}C_4{\alpha }^4=7315$

$\Rightarrow \left(\frac{22!}{4!\times 18!}\right){\alpha }^4=7315$

$\Rightarrow 7315{\alpha }^4=7315$

$\Rightarrow {\alpha }^4=1$

$\Rightarrow \left|\alpha \right|=1$