Question

Is $x^5$ in the expansion of ${[2x^3-\frac{1}{3x^3}]}^{10}$.

Answer 1

${[2x^3-\frac{1}{3x^3}]}^{10}$

$T_{r+1}={}^{10}{C}_r(2x^3{)}^{10-r}{\left(\frac{-1}{3x^3}\right)}^r$

$={}^{10}{C}_r{2}^{10-r}.x^{30-3r-3r}{(-\frac{1}{3})}^r$

$={}^{10}{C}_r{.2}^{10-r}.x^{30-6r}{(-\frac{1}{3})}^r$

We need $x^5$ term so put

$30-6r=5$

$6r=25$

$r=\frac{25}{r}$

it can't be fraction

So, $x^5$ is not define in the expansion.