If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{l}}\right)^{9}$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^{\alpha} \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to ________.

Given binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{l}}\right)^{9}$ <br/><br/>$$ \begin{aligned} T_{r+1} & ={ }^{9} C_{r}\left(\frac{x^{\frac{5}{2}}}{2}\right)^{9-r}\left(\frac{-4}{x^{l}}\right)^{r} \\ & ={ }^{9} C_{r} x^{\frac{45-5 r}{2}-lr} \cdot 2^{r-9} \cdot 4^{r} \cdot(-1)^{r} \end{aligned} $$ <br/><br/>For constant term, power of x is zero. <br/><br/>So, $\frac{45-5 r}{2}=l r \Rightarrow 2 l r+5 r=45$ <br/><br/>Now constant term $=-84$ <br/><br/>and ${ }^{9} C_{r} \cdot 2^{3 r-9}(-1)^{r}=-84$ <br/><br/>So, $r=3$ and $l=5$ <br/><br/>Now for $x^{-15}, \frac{45-5 r}{2}-5 r=-15$ <br/><br/>$\Rightarrow$ $45-15 r=-30$ <br/><br/>$\Rightarrow$ $r=5$ <br/><br/>$\therefore $ Coefficient $=-{ }^{9} C_{5} 2^{6}=-63.2^{7}$ <br/><br/>$\therefore \alpha=7, \beta=-63$ <br/><br/>and $|\alpha l-\beta|=|7 \times 5+63|=98$