Find the term independent of x, where `x =! 0`, in the expansion of `((3x^(2))/(2) - (1)/(3x))^(15)`.

Explanation

Let `T_(r+1)` be independent of x.
Here, `T_(r+1) = (-1)^(r) xx.^(15) C _(r) xx ((3x^(2))/2)^((15-r))xx (1/(3x))^(r)`
`rArr T_(r+1)=(-1)^(r) xx.^(15)C_(r) xx 3 ^((15-2r)) xx 2 ^((r-15)) xxx^((30-3r))." "` …(i)
Now, `T_(r+1)` will be independent of x only when the power of x in it is 0.
`:. 30 - 3r = 0 rArr 3r = 30 rArr r = 10 rArr r+1 = 11.`
Thus , `T_(11)` is independent of x.
Now, `T_(11)=T_((10+1))`
`=(-1)^(10 ) xx .15 C_(10) xx 3 ^ ((15-20)) xx 2 ^(10-15) xx x^(0) ` [putting r = 10 in(i) ]
`.^(15)C_(5) xx 3 ^(-5) xx 2^(-5) = (.^(15)C_(5))/(6^(5)) " "[ :' .^(15) C _(10) = .^(15) C _((15-10)) = .^(15) C _(5)]`
`=((15 xx 14 xx 13 xx 12 xx 11)/(5 xx 4 xx 3 xx 2 xx 1 ) xx1/(36 xx 36 xx 6)) = 1001/2592.`
Hence, the term independent of x in the given expansion is `1001/2592`