Find the term independent of x in the expansion of : (2x+1/3x^2)^9

Question

Find the term independent of x in the expansion of : \(\Big(2x+\frac{1}{3x^2}\Big)^9\)

Answer 1

For \(\Big(2x+\frac{1}{3x^2}\Big)^9\)

a = 2x, b = \(\frac{1}{3x^2}\) and n = 9

We have a formula,

Now, to get coefficient of term independent of x that is coefficient of x⁰ we must have, 

(x)^9-3r = x⁰ 

• 9 - 3r = 0 

• 3r = 9 

• r = 3

Therefore, coefficient of x⁰ = (⁹₃) \(\frac{(2)^{9-3}}{(3)^3}\) 

= \(\frac{9\times8\times7(2)^6}{3\times2\times1(3)^3}\) 

= \(\frac{1792}{3}\) 

Conclusion : coefficient of x⁰ = \(\frac{1792}{3}\)