Simplification and simultaneous equations...... I don't need help though...... I made this question up!! :)

2) x = -1, y = 3 (Real solutions)

    x = -1+(2 i π c_2)/(log(3)) and y = (log(64)+2 i π c_1)/(log(4)) and c_1 element Z and c_2 element Z(Complex solutions).

 

1) MAX: THIS WILL MAKE YOUR HEAD SPIN!!!.

Expand the following:
(1/x-1/y)/((π x-π y) (x^2+(π x y)^(1/9)+8 π y^2))

(x^2+8 π y^2+π^(1/9) (x y)^(1/9)) (π x-π y) = π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)):
(1/x-1/y)/(π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))

(1/x-1/y)/(π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))) = 1/((π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))) x)-(1)/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))):
1/(x (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

x (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))) = x π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))+x (-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))):
1/π x x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

x π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9)) = x^2 π (x^2+8 π y^2+π^(1/9) (x y)^(1/9)):
1/(π x^2 (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

π x^2 (x^2+8 π y^2+π^(1/9) (x y)^(1/9)) = π x^2 x^2+π x^2×8 π y^2+π x^2 π^(1/9) (x y)^(1/9):
1/(π x^2 x^2+π×8 π x^2 y^2+π π^(1/9) x^2 (x y)^(1/9)-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

π x^2 x^2 = π x^4:
1/(π x^4+π×8 π x^2 y^2+π π^(1/9) x^2 (x y)^(1/9)-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

π x^2×8 π y^2 = π^2 x^2×8 y^2:
1/(π x^4+π^2×8 x^2 y^2+π π^(1/9) x^2 (x y)^(1/9)-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

π x^2 π^(1/9) (x y)^(1/9) = π^(1+1/9) x^2 (x y)^(1/9):
1/(π x^4+π^2×8 x^2 y^2+π^(1+1/9) x^2 (x y)^(1/9)-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

1+1/9 = 10/9:
1/(π x^4+π^2×8 x^2 y^2+π^10/9 x^2 (x y)^(1/9)-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

-π x y (x^2+8 π y^2+π^(1/9) (x y)^(1/9)) = -(π x y x^2)-π x y×8 π y^2-π x y π^(1/9) (x y)^(1/9):
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)+-(π x y x^2)-π×8 π x y y^2-π π^(1/9) x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

-π x y x^2 = -π x^(1+2) y:
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)+-(π x^(1+2) y)-π×8 π x y y^2-π π^(1/9) x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

1+2 = 3:
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)-π x^3 y-π×8 π x y y^2-π π^(1/9) x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

-π x y×8 π y^2 = -π^2 x y^(1+2)×8:
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)-π x^3 y+-(π^2×8 x y^(1+2))-π π^(1/9) x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

1+2 = 3:
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)-π x^3 y-π^2×8 x y^3-π π^(1/9) x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

-π x y π^(1/9) (x y)^(1/9) = -π^(1+1/9) x y (x y)^(1/9):
1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)-π x^3 y-π^2×8 x y^3+-(π^(1+1/9) x y (x y)^(1/9)))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

1+1/9 = 10/9:
Answer: | 1/(π x^4+π^2×8 x^2 y^2+π^(10/9) x^2 (x y)^(1/9)-π x^3 y-π^2×8 x y^3-π^10/9 x y (x y)^(1/9))-1/(y (π x (x^2+8 π y^2+π^(1/9) (x y)^(1/9))-π y (x^2+8 π y^2+π^(1/9) (x y)^(1/9))))

LOL Guest #1, you need not do that to me in question #1.

\(\dfrac{(x^{-1}-y^{-1})}{(\pi x-\pi y)(x^2+9\sqrt{\pi}xy+8\pi y^2)}\\ =\dfrac{\frac{y-x}{xy}}{\pi(x-y)(x+\sqrt{\pi}y)(x+8\sqrt{\pi}y)}\\ =-\dfrac{\frac{1}{xy}}{\pi (x+\sqrt{\pi}y)(x+8\sqrt{\pi}y)}\\ = -\dfrac{1}{\pi xy (x+\sqrt{\pi}y)(x+8\sqrt{\pi}y)}\)

You are complexifying that question(Yeah you never heard that word..... I made it up!! :) )