Epic Factoring

Let $x = \sqrt{1+ (50 \times 51 \times 52 \times 53 ) }$ . Then, $x^{2} = 1+ (50 \times 51 \times 52 \times 53 )$ . Notice here that we can subtract the $1$ to evoke a difference of squares on the left hand side, which would prove useful. We do this to obtain

$x^{2} -1 = (50 \times 51 \times 52 \times 53 )$ .

Factoring the left side, we are left with $(x+1)(x-1) = (50 \times 51 \times 52 \times 53 )$

This tells us that the product on the right hand side is a product of two numbers that differ by $2$ . The closest we can pair the numbers up on the right is to do so with the pairs $53 \times 50$ and $52 \times 51$ . Note that these equal $2650$ and $2652$ respectively, so we conclude that $x = \boxed{2651}$ .