Find the term independent of a in the expansion of `(a^((1)/(3))+(1)/(2a^((1)/(3))))^(18),a gt0`.

Explanation

We have,
`T_(r+1)=.^(18)C_(r)(a^((1)/(3)))^(18-r)((1)/(2a^((1)/(3))))^(r)`
`.^(18)C_(r)a^((18-r)/(3))*((1)/(2^(r)))*(1)/(a^((r)/(3)))=.^(18)C_(r)*(1)/(2r)a^(18-2r)/(3)`
Since we want to find the term independence of a, so, the exponent of a must be equal to zero
`i.e., (18-2r)/(3)=0`
`impliesr=9`
hence, the required terms is `.^(18)C_(9)((1)/(2))^(9)`