Question

Find the independent term in the expansion of $(5x^2-\frac{1}{x^4}{)}^6$.

Answer 1

In the expansion $(a{x}^p+\frac{b}{x^q}{)}^n$, the independent term is $T_{r+1}$ where $r=\frac{np}{p+q}$.

Here, $r=\frac{6\times 2}{6}=2$.

The independent term is $T_3$.

$T_3=T_{2+1}={6_C}_2\left(5x^2{)}^4\right(\frac{-1}{x^4}{)}^2$

$=5^4\times 15=9375$.