A large root!

In general,

$\sum_{n=1}^{N}n^3 = (\sum_{n=1}^{N}n)^2 = (\frac{N(N+1)}{2})^2$

So,

$\sqrt{\sum_{n=1}^{N}n^3} = \frac{N(N+1)}{2}$

And for N = 1000000,

$\sqrt{\large \sum_{n=1}^{1000000}n^3} = \boxed{500000500000}$