Answer
$-84$
Work Step by Step
We are given the expression:
$\left(x-\dfrac{1}{x^2}\right)^9$
The term $T_{k+1}$ of the expansion of $(a+b)^n$ is:
$T_{k+1}=\binom{n}{k}a^{n-k}b^r$
We have:
$T_{k+1}=\binom{9}{k}x^{9-k}(-x^{-2})^k=(-1)^k\binom{9}{k}x^{9-k-2k}=(-1)^k\binom{9}{k}x^{9-3k}$
Determine $k$ so that the term contains $x^0$:
$9-3k=0$
$9=3k$
$k=3$
Determine the coefficient of $x^0$:
$(-1)^3\binom{9}{3}=-\dfrac{9!}{3!6!}=-84$
