Q.Find the term independent of $x$ in the expansion of $\left(x^2 - \dfrac{2}{x}\right)^{9}$.

A term independent of $x$ is the specific term whose net power of $x$ works out to zero, found by setting the general term's exponent to zero and solving for $r$.

The general term of $\left(x^2+\left(-\dfrac{2}{x}\right)\right)^9$ is

The term independent of $x$ occurs where the exponent of $x$ is zero: $18-3r=0 \Rightarrow r=6$.

So the required term is $T_7$ (since $r+1=7$):

$^{9}C_{6} = \,^{9}C_{3}$ (by the symmetry property, since $9-6=3$) $= \dfrac{9!}{3!6!} = \dfrac{9\times8\times7}{3\times2\times1} = 84$.

$T_7 = 84\times64 = 5376$ …

Errors combining the exponents of $x$ from the two factors — $(x^2)^{9-r}$ contributes $x^{18-2r}$ and $\left(\dfrac{1}{x}\right)^r$ contributes $x^{-r}$, and these two must be added (not confused) to get the net exponent $18-3r$; also forgetting the sign, since $\left(-\ …