Do We Multiply Or Add?

Relevant wiki: Sum of n, n², or n³

$a^1 \times a^2 \times a^3 \times \ldots \times a^n = a^{276}\\ a^{1+2+3+\ldots+n} = a^{276}\\ 1+2+3+\ldots+n = 276\\ \displaystyle \sum_{i=1}^n i = 276\\ \dfrac{n(n+1)}{2} = 276\\ n^2+n = 552\\ n^2 + n -552=0\\ (n+24)(n-23) = 0\\ n=-24,\;23$

We know that $n > 0$ , therefore the answer is $n=\boxed{23}$

We first note that, $\large a^m\times a^n = a^{m+n}$

$\large \implies a^1\times a^2\times\times...\times a^{23}$

$\large = a^{\displaystyle \sum_{n=1}^{23} n } = a^{\frac{23\times24}{2}} = a^{276}$

$\large \implies n= \boxed{23}$