The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is ______.

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Binomial Theorem

The number of rational terms in the binomial expansion of

(4^{\frac{1}{4}} + 5^{\frac{1}{6}})^{120}

is ______.

Answer: 21

Since

4^{1/4} = 2^{1/2}

, the general term of the expansion is

T_{r + 1} =^{120} C_{r} (2^{1/2})^{120 - r} (5^{1/6})^{r} =^{120} C_{r} 2^{\frac{120 - r}{2}} 5^{\frac{r}{6}}

A term is rational exactly when both exponents are integers, which needs

120 - r

even and

r

divisible by

6

. Every multiple of

6

is already even, so the two conditions collapse to

6 ∣ r

with

0 \leq r \leq 120

, giving

r = 0, 6, 12, \dots, 120

. Counting these values of

r

, the expansion has

21

rational terms.

JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Integer answer