Find all of the solutions of the equation x4-625i=0 and represent the solutions graphically.

what is the next step to answer the question?
(x^2-25√i)(x^2+25√i)
(x-5∜i)(x+5∜i)(x-5∜i i)(x+5∜i i)
i = 1 cis π/2
∜i = ∜1 cis π/8

x^4=625@(90+N360)

x^2=+-25@(45+n180)
x=5@(22.5+n90)

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you can convert these to complex numbers if you wish, but I wouldn't.

Graphically: plot it polar, real axis to the right, imag upward, then draw rays of length 5 at the indicated angles.
check my work.

The next step to answer the question is to find the roots of the equation x^4 - 625i = 0.

To do this, we can rewrite the equation as a product of two factors:

(x^2-25√i)(x^2+25√i)

Now, let's factor each factor further:

x^2-25√i = (x-5∜i)(x+5∜i)

x^2+25√i = (x-5∜i i)(x+5∜i i)

Next, we can find the roots of each factor individually:

For the factor (x-5∜i), the root is x = 5∜i

For the factor (x+5∜i), the root is x = -5∜i

For the factor (x-5∜i i), we can use the fact that ∜i = ∜1 cis π/8. Therefore, the root is x = 5∜1 cis π/8 = 5cis(π/8)

For the factor (x+5∜i i), the root is x = -5∜1 cis π/8 = -5cis(π/8)

So, the solutions to the equation x^4 - 625i = 0 are:

x = 5∜i, -5∜i, 5cis(π/8), -5cis(π/8)

To represent these solutions graphically, we can plot them on the complex plane as points. The point 5∜i would be located at 5 units away from the origin in the direction of ∜i (angle of π/8), and the point -5∜i would be located at the same distance but in the opposite direction. The points 5cis(π/8) and -5cis(π/8) would be represented respectively by 5 units away from the origin in the direction of the angles π/8 and -π/8.

To find the solutions of the equation x^4 - 625i = 0, we can start by factoring the equation.

The given equation can be written as (x^2)^2 - (25√i)^2 = 0.

Using the difference of squares identity, we can factor this as (x^2 - 25√i)(x^2 + 25√i) = 0.

Now, we have two quadratic equations: x^2 - 25√i = 0 and x^2 + 25√i = 0.

Let's solve the first equation:

x^2 - 25√i = 0
Taking the square root of both sides, we get:
x = ± 5√√i

Since √i = √(1 cis π/2) = 1 cis (π/2)/2 = 1 cis π/4, we can simplify the solutions further:
x = ± 5√(1 cis π/4)

To fully represent these solutions graphically, we need to convert them to the complex number form, where a complex number is written as a + bi.

Using the cis (polar) form of complex numbers, we can rewrite the solution as:

x = ± 5√(cos(π/4) + i sin(π/4))

Using the identity cos(π/4) = sin(π/4) = √2/2, we have:

x = ± 5(√2/2 + i√2/2)

Expanding this expression further, we get:

x = ± (5√2/2 + 5i√2/2)

Therefore, the solutions to the first equation are:
x = (5√2/2 + 5i√2/2) and x = (-5√2/2 - 5i√2/2).

Similarly, we can solve the second equation:

x^2 + 25√i = 0
Taking the square root of both sides, we get:
x = ± 5i√√i

Similar to before, we can simplify the solutions further using cis (polar) form:
x = ± 5i√(1 cis π/8)

Expanding this expression further using the identity cos(π/8) = sin(π/8) = (√2+√6)/4, we have:

x = ± 5i(√2+√6)/4

Therefore, the solutions to the second equation are:
x = (5i(√2+√6))/4 and x = (-5i(√2+√6))/4.

By representing these solutions graphically, we can plot the complex numbers on the complex plane. The real part of the complex number is plotted along the x-axis, and the imaginary part is plotted along the y-axis.

For the solutions x = (5√2/2 + 5i√2/2) and x = (-5√2/2 - 5i√2/2), we plot the points (5√2/2, 5√2/2) and (-5√2/2, -5√2/2) respectively.

For the solutions x = (5i(√2+√6))/4 and x = (-5i(√2+√6))/4, we plot the points (0, 5(√2+√6)/4) and (0, -5(√2+√6)/4) respectively.

These points on the complex plane represent the solutions to the given equation x^4 - 625i = 0.