Solve for z:
(z-5)^4+(z-3)^4 = -8
Expand out terms of the left hand side:
2 z^4-32 z^3+204 z^2-608 z+706 = -8
Add 8 to both sides:
2 z^4-32 z^3+204 z^2-608 z+714 = 0
The left hand side factors into a product with three terms:
2 (z^2-8 z+17) (z^2-8 z+21) = 0
Divide both sides by 2:
(z^2-8 z+17) (z^2-8 z+21) = 0
Split into two equations:
z^2-8 z+17 = 0 or z^2-8 z+21 = 0
Subtract 17 from both sides:
z^2-8 z = -17 or z^2-8 z+21 = 0
Add 16 to both sides:
z^2-8 z+16 = -1 or z^2-8 z+21 = 0
Write the left hand side as a square:
(z-4)^2 = -1 or z^2-8 z+21 = 0
Take the square root of both sides:
z-4 = i or z-4 = -i or z^2-8 z+21 = 0
Add 4 to both sides:
z = 4+i or z-4 = -i or z^2-8 z+21 = 0
Add 4 to both sides:
z = 4+i or z = 4-i or z^2-8 z+21 = 0
Subtract 21 from both sides:
z = 4+i or z = 4-i or z^2-8 z = -21
Add 16 to both sides:
z = 4+i or z = 4-i or z^2-8 z+16 = -5
Write the left hand side as a square:
z = 4+i or z = 4-i or (z-4)^2 = -5
Take the square root of both sides:
z = 4+i or z = 4-i or z-4 = i sqrt(5) or z-4 = -i sqrt(5)
Add 4 to both sides:
z = 4+i or z = 4-i or z = 4+i sqrt(5) or z-4 = -i sqrt(5)
Add 4 to both sides:
Answer: |z = 4+i or z = 4-i or z = 4+i sqrt(5) or z = 4-i sqrt(5)