Each of the 1010 terms of the product can be expressed as sqrt(2)^(2x-1) + sqrt(2)^(2x) for x=1 to 1010.

The expression can be simplified to 2^(x-1) * (sqrt(2) + 2).

Then for all 1010 terms the product is (2^0 * 2^1 * ... * 2^1009) * (sqrt(2) + 2)^1010 = 2^509545 * sqrt(2)^1010 * (1+sqrt(2))^1010

Some logs yields 1+sqrt(2) = 2^1.27155. Then the product simplifies to 2^(509545 + 505 + 1284.26884) = 2^511334.26884. Then the value k sought is **511334**