Solve the equation finding all solutions. Write all answers in trigonometric and in exact standard form.

Question

x^4+81=0

Answer 1

x^4 + 81 = 0

x^4 = -81

x^4 = 81e^πi = 81e^3πi = 81e^5πi = 81e^7πi

x = 3 (cos π/4 + i sin π/4)

x = 3 (cos 3π/4 + i sin 3π/4)

x = 3 (cos 5π/4 + i sin 5π/4)

x = 3 (cos 7π/4 + i sin 7π/4)

or

x = 3 (cos 45° + i sin 45°)

x = 3 (cos 135° + i sin 135°)

x = 3 (cos 225° + i sin 225°)

x = 3 (cos 315° + i sin 315°)

Changing to Standard Form,

x = 3 (√2/2 + √2/2i) = 3/2 √2 + 3/2 √2 i

x = 3 (-√2/2 + √2/2i) = -3/2 √2 + 3/2 √2 i

x = 3 (-√2/2 - √2/2i) = -3/2 √2 - 3/2 √2 i

x = 3 (√2/2 - √2/2i) = 3/2 √2 - 3/2 √2 i