Question

The term independent of x in the expansion of ${(x^2-\frac{3\sqrt{3}}{x^3})}^{10}$ is

Answer 1

An expression consisting of two terms, connected by $+$ or $–$ sign is called a binomial expression.

$(r-1{)}^{th}$ term be independent of x which is given by:

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

${=}^{10}{C}_r$ $x^{2(10-r)}$ $3\ast 3^{r/2}\times \frac{1}{x^{3r}}$

${=}^{10}{C}_r$ $\frac{x^{20-2r}}{x^{3r}}$ $3\times 3^{r/2}$

${=}^{10}{C}_r$ $x^{20-5r}$ $3\times 3^{r/2}$

since the term is independent of x we have:

20 - 5r = 0 => r= 4.

Hence, third term is independent of x and its value is given by:

$T_3=10C_{4}3\times 3^{4/2}=27$