Animated Solution for Mathematics - Binomial Theorem: If the number of integral terms in the expansion of $(\sqrt[3]{3}+\sqrt[5]{5}{)}^n$ is exactly 33, then the least value of $n$ is:

We are tasked with finding the least value of $n$ such that the expansion of $(\sqrt[3]{3}+\sqrt[5]{5}{)}^n$ contains exactly 33 integral terms.

Note that the expression provided in the prompt contains a slight discrepancy in the roots. Based on the logic of the problem, we define the expression as:

The General Term

In any binomial expansion $(a+b{)}^n$, the general term is given by:

Substituting $a=3^{1/3}$ and $b=5^{1/5}$, the general term becomes:

The Exponents

For a term to be an integer, the exponents of the prime bases 3 and 5 must be non-negative integers. This leads to two conditions:

1. The exponent of 5, $\frac{r}{5}$, must be an integer. This implies $r$ must be a multiple of 5, i.e., $r\in \{0,5,10,\dots \}$.

2. The exponent of 3, $\frac{n-r}{3}$, must be an integer. This implies $(n-r)$ must be a multiple of 3.

Finding the Pattern

Since $r$ must be a multiple of 5, we have $r=5k$ for $k=0,1,2,\dots$. Substituting this into the second condition, we require:

For a fixed $n$, the values of $r$ that yield integral terms are those where $r$ is a multiple of 5 and $r\equiv n(\mathrm{mod}3)$. Since 5 and 3 are coprime, the values of $r$ will form an arithmetic progression with a common difference of $\text{lcm}(3,5)=15$.

The valid values of $r$ are $r_0,r_0+15,r_0+30,\dots ,r_0+15(N-1)$, where $N$ is the number of terms. We are given $N=33$.

The 33rd term is given by $r_{33}=r_0+(33-1)\times 15=r_0+32\times 15=r_0+480$.

Since $r$ must satisfy $0\le r\le n$, the smallest possible value for the 33rd term occurs when $r_0$ is the smallest non-negative integer satisfying the divisibility conditions. For $n$ to be the least value, we set $r_0=0$ (which implies $n$ must be a multiple of 3).

Thus, the 33rd term index is $r=480$. To ensure this is the last term, we require $n\ge 480$.

The next potential integral term would occur at $r=480+15=495$. To ensure there are exactly 33 terms, we must have $n<495$.

The least value of $n$ that satisfies the condition is 480.