Find the term independent of $x$ in the expansion of
${(x^{2} - \frac{2}{x^{3}})}^{5}$

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Solution

$(x^{2} - \frac{2}{x^{3}})^{5}$
$T_{r + 1} =^{5} C_{r} (x^{2})^{5 - r} {(\frac{- 2}{x^{3}})}^{r}$
$=^{5} C_{r} (- 2)^{r} x^{10 - 2 r - 3 r}$
$^{5} C_{r} (- 2)^{r} x^{10 - 5 r}$
$10 - 5 r = 0$
$∴ r = 2$
$T_{3} =^{5} C_{2} (- 2)^{2}$
$= \frac{120}{2 \times 6} \times 4$
$= 40$

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