Animated Solution for Mathematics - Binomial Theorem: If $n$ is the number of irrational terms in the expansion of $(3^{1/4}+5^{1/8}{)}^{60}$, then $(n-1)$ is divisible by:

Imagine you are standing before the expression $(3^{\frac{1}{4}}+5^{\frac{1}{8}}{)}^{60}$. It looks daunting, but in the world of JEE Advanced, we use the elegance of the Binomial Theorem rather than brute force.

Our mission is to find $n$, the number of irrational terms, and then solve a final divisibility puzzle.

Every term in a binomial expansion follows the general term formula:

Here, our $a=3^{\frac{1}{4}}$, our $b=5^{\frac{1}{8}}$, and our $n=60$. Plugging these into the formula, we get:

Using the laws of indices, we simplify this to:

A term is rational if and only if its exponents are integers. If the exponents are fractions, the term involves roots, making it irrational.

We require both $\frac{60-r}{4}\in \mathbb{Z}$ and $\frac{r}{8}\in \mathbb{Z}$.

Starting with the second condition, $\frac{r}{8}\in \mathbb{Z}$ implies that $r$ must be a multiple of $8$. Given the constraint $0\le r\le 60$, our candidates for $r$ are:

Now, we check the first condition: $\frac{60-r}{4}\in \mathbb{Z}$. Since $60$ is divisible by $4$, we simply need $r$ to be divisible by $4$.

Because all our candidates are multiples of $8$, they are automatically multiples of $4$. Thus, all $8$ values of $r$ produce rational terms.

The total number of terms in the expansion is $n+1=60+1=61$.

Since there are $8$ rational terms, the number of irrational terms $n$ is:

The question asks for the value of $(n-1)$. Therefore: