Coefficient of $x^5$ in $(4x^2- \frac{1}{x^3})^8$ I took one $x$ out of bracket, and I got $\frac{1}{x^{24}}  (4x^5-1)^8.$ Now, we get $x$ to the power of if we get $x$ to the power $29,$ then we divide it by $24,$ we get $x$ to the power $5,$ but i can't get $29,$ by multiplying $5.$

The term in given expansion containing $x^5$ $=^8C_r(4x^2)^r(-1/x^3)^{8-r} = ^8C_r4^r(-1)^{8-r}x^{2r-3(8-r)} = ^8C_r4^r(-1)^{8-r}x^{5r-24}$
For $x^5$, $5r-24=5$, so $r=29/5$ which is not a whole number. So there is no term in the expansion of $(4x^2-(1/x^3))^8$ having term with $x^5$. So coefficient of $x^5$ in the given expansion is $\color{red}{0}$
