It is easy to show x^4+y^4=16 is contained in the circle x^2+y^2=4*sqrt(2). But x+y=4 is tangent to the circle x^2+y^2=8 . Since 8 > 4*sqrt(2) , then there is no intersection.

Conclusion: There is no "real solution" to your question, only "imaginary solution"!