What is  the value of $$x^4 + y^4$$ when the value of $$x^3+y^3 = 8$$ and $$x + y = 2$$?

Given : $$x^3+y^3=8$$ -----------(i)

and $$x+y=2$$ ------------(ii)

Cubing both sides, we get :

=> $$(x+y)^3=(2)^3$$

=> $$x^3+y^3+3xy(x+y)=8$$

Substituting values from equations (i) and (ii),

=> $$8+3xy(2)=8$$

=> $$6xy=8-8=0$$

=> $$xy=0$$ -----------(iii)

Now, squaring equation (ii), => $$(x+y)^2=(2)^2$$

=> $$x^2+y^2+2xy=4$$

=> $$x^2+y^2=4$$      $$[\because xy=0]$$

Similarly, again squaring both sides, we get :

=> $$x^4+y^4+2x^2y^2=16$$

=> $$x^4+y^4+2(xy)^2=16$$

=> $$x^4+y^4=16$$

=> Ans - (C)