To solve this system of equations, we can start by isolating x and y in the first equation:

log√2 (x+y) = 2

Rewrite the equation using the property of logarithms:

√2^2 = x + y

Simplify:

2 = x + y

Now we can use this simplified equation to substitute x in the second equation:

3^x * 7^x = 21

Rewrite using exponents:

(3*7)^x = 21

Simplify:

21^x = 21

Since 21 = 3 * 7, we can rewrite the equation as:

(37)^x = 37

This gives us:

x = 1

Substitute x=1 back into the simplified equation we found earlier:

2 = x + y
2 = 1 + y
y = 1

Therefore, the solution to the system of equations is x = 1, y = 1.