JEE Main 2021 (Online) 25th July Evening ShiftQuestion 1
The sum of all those terms which are rational numbers in the expansion of $\left(2^{1/3}+3^{1/4}\right)^{12}$ is :
Binomial Theorem · Rational and Integral Terms
Solution
Concept. A term is rational when both exponents are whole numbers. Formulas used. (1) $T_{r+1}={}^{n}C_r\,a^{n-r}b^r$Step 1 - General term. $$T_{r+1}={}^{12}C_r\,2^{\frac{12-r}{3}}\,3^{\frac r4}.$$ Step 2 - Conditions. $3\mid(12-r)$ means $3\mid r$; with $4\mid r$ this gives $12\mid r$, so $r=0$ or $12$. Step 3 - Sum. $r=0$: $2^4=16$; $r=12$: $3^3=27$. Total $43$. Answer: (C)