The term independent of x in the expansion of ((1)/(60) - x ^ 8 81...

To find the coefficient of term independent of $x$ in given expression, first let's find constant term and coefficient of ${x}^{- 8} in the expansion of${\left(2{x}^{2} - \dfrac{3}{{x}^{2}}\right)}^{6}$<br />${T}_{r + 1}{=}^{6}{C}_{r}{\left(2{x}^{2}\right)}^{6 - r} \times {\left(- \dfrac{3}{{x}^{2}}\right)}^{r}$<br />${T}_{r + 1}{=}^{6}{C}_{r}{2}^{6 - r} \times {\left(- 3\right)}^{r} \times {x}^{12 - 4r}$<br />For constant term:$r = 3$, hence constant term =${}^{6}{C}_{3}{2}^{3} \times {\left(- 3\right)}^{3} \times {x}^{0}$<br />For coefficient of${x}^{- 8}: r = 5$, hence coefficient of${x}^{- 8}{=}^{6}{C}_{5} \times {2}^{1}{\left(- 3\right)}^{5} \times {x}^{- 8}$<br />Given expression can be written as<br />\left(\dfrac{1}{60} - \dfrac{{x}^{8}}{81}\right)\left({}^{6}{C}_{3} \times {2}^{3} \times {\left(- 3\right)}^{3} \times {x}^{0} + \ldots + {}^{6}{C}_{5} \times {2}^{1} \times {\left(- 3\right)}^{5} \times {x}^{- 8} + \ldots\right)$
Hence, coefficient of term independent of $x$
$= \dfrac{1}{60}{\times}^{6}{C}_{3} \times {2}^{3}{\left(- 3\right)}^{3} - \dfrac{1}{81}{\times}^{6}{C}_{5} \times {2}^{1} \times {\left(- 3\right)}^{5}$
$= - 72 + 36 = - 36$