Question

If the term without $x$ in the expansion of $\left(x^{\frac{2}{3}}+\frac{\alpha}{x^3}\right)^{22}$ is $7315$ , then $|\alpha|$ is equal to ___

Answer 1

$T _{ r +1}={ }^{22} C _{ r } \cdot\left( x ^{\frac{2}{3}}\right)^{22- r } \cdot(\alpha)^{ r }, x ^{-3 r } $
$ ={ }^{22} C _{ r } \cdot x ^{\frac{44}{3}-\frac{2 r }{3}-3 r }(\alpha)^{ r } $
$ \frac{44}{3}=\frac{11 r }{3} $
$ r =4 $
${ }^{22} C _4 \cdot \alpha^4=7315 $
$ \frac{22 \times 21 \times 20 \times 19}{24} \cdot \alpha^4=7315$
$\alpha=1$