Animated Solution for Mathematics - Binomial Theorem: The number of rational terms in the binomial expansion of $(4^{\frac{1}{4}}+5^{\frac{1}{6}}{)}^{120}$ is ___

The binomial theorem states that any term in the expansion of $(a+b{)}^n$ is given by the general term formula:

In this problem, we are given the expression $(4^{1/4}+5^{1/6}{)}^{120}$. Here, $a=4^{1/4}$, $b=5^{1/6}$, and $n=120$.

A common trap in JEE problems is failing to reduce bases to their prime forms. We must express $4$ as $2^2$:

Substituting this into our general term formula, we get:

Simplifying the exponents, the expression becomes:

For the term $T_{r+1}$ to be rational, the exponents of the prime bases $2$ and $5$ must be integers. This yields two simultaneous conditions:

1. $\frac{120-r}{2}\in \mathbb{Z}⟹r$ must be even. 2. $\frac{r}{6}\in \mathbb{Z}⟹r$ must be a multiple of $6$.

Since any multiple of $6$ is inherently an even number, the condition simplifies to $r$ being a multiple of $6$. Given that $0\le r\le 120$, the possible values for $r$ are $0,6,12,\dots ,120$.

The values of $r$ form an arithmetic progression where the first term $a=0$, the common difference $d=6$, and the last term $l=120$. The number of terms $N$ is calculated as:

The total number of rational terms in the expansion is 21.