smallest positive integer nt

Question

Let $a$ be the smallest positive integer such that $a*b$ is a perfect $n$th power of an integer for some $n \ge 2$, where $b=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{7^3}$. What is $a+n$?

So I noticed that $343$ and $1694$ are divisible by 7 so I'm thinking this might be a $n$ could be $7$ but I'm not sure how to find $a$ and $b$ then.

Answer 1

$a = 2^13^1 =6$ works because $1980 + 1, 384 + 1, 1694$, and $7^3$ are all divisible by $7$, meaning $a \times b$ can be written as $x^7$.

It's trivial to rule out $a < 6$.