Question

Find the constant term in the exp ${(4x^2-\frac{3}{2x})}^{12}$.

Answer 1

Given expression ${(4x^2-\frac{3}{2x})}^{12}$

$t_{r+1}{=}^n{C}_r{x}^r{y}^{n-r}$

$t_{r+1}{=}^{12}{C}_r{\left(4x^2\right)}^r\cdot {\left(\frac{-3}{2x}\right)}^{n-r}$

$=12c_r{4}^r\cdot x^{2r}\cdot \frac{(-3{)}^{12-r}}{2^{12-8}}\cdot \frac{1}{(x{)}^{12-r}}$

$\Rightarrow 2r=12-r$

$\Rightarrow r=4$

$\begin{array}{cc}\Rightarrow t_{r+1}=t_5& =12c_4\cdot 4^4\cdot \frac{(-3{)}^8}{2^8}\\ & =\frac{12\times 11\times 10\times 9}{4\times 3\times 2\times 1}\times 4^4\cdot \frac{\times 3^8}{2^8}\end{array}$

$=3247695$