Find All Complex Number Solutions x^4+i=0

$x^{4} + i = 0$

Subtract $i$ from both sides of the equation.

$x^{4} = - i$

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Substitute the actual values of $a = 0$ and $b = - 1$ .

$| z | = \sqrt{{(- 1)}^{2}}$

Find $| z |$ .

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1}$

Any root of $1$ is $1$ .

$| z | = 1$

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- 1}{0})$

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = 1$ .

$\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2})$

Replace the right side of the equation with the trigonometric form.

$x^{4} = \cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2})$