Find all of the solutions of the equation x^4-625i=0 and represent the solutions graphically.
(x^2-25√i)(x^2+25√i)
(x-5∜i)(x+5∜i)(x-5∜i i)(x+5∜i i)
Now just use de Moivre's formulas to find ∜i.
If you want, you can rephrase these solutions as
±5(-1)^(1/8)
±5(-1)^(5/8)
To find all the solutions of the equation x^4 - 625i = 0, we can start by factoring the equation.
Let's rewrite the equation as x^4 = 625i.
Taking the fourth root of both sides, we get:
x = (625i)^(1/4)
Now, let's express 625 in terms of its modulus and argument:
625 = 625(1) = 625e^(0i),
where modulus |625| = 625 and argument arg(625) = 0.
Now, we can express 625 in polar form:
625 = 625e^(0i) = 625e^(2πni), where n is an integer
Taking the fourth root of 625 in polar form, we get:
x = (625e^(2πni))^(1/4)
Using the exponential property, we can simplify this expression:
x = (625)^(1/4) * (e^(2πi * n/4))^(1/4)
Simplifying further, we have:
x = 5 * (e^(πi * n/2)), where n = 0, 1, 2, 3
Now, let's substitute the values of n in the above expression to find the solutions:
When n = 0, x = 5 * (e^(πi * 0/2)) = 5 * e^0 = 5
When n = 1, x = 5 * (e^(πi * 1/2)) = 5 * e^(πi/2) = 5i
When n = 2, x = 5 * (e^(πi * 2/2)) = 5 * e^(πi) = -5
When n = 3, x = 5 * (e^(πi * 3/2)) = 5 * e^(3πi/2) = -5i
Therefore, the solutions to the equation x^4 - 625i = 0 are x = 5, 5i, -5, and -5i.
To represent these solutions graphically, we can plot them on the complex plane. The points corresponding to the solutions are (5, 0), (0, 5), (-5, 0), and (0, -5).
To find the solutions of the equation x^4 - 625i = 0, we can start by rearranging the equation:
x^4 = 625i.
Now, we need to express 625i in polar form. The polar form of a complex number is given by r(cosθ + isinθ), where r is the magnitude or absolute value of the complex number, and θ is the angle in radians that the complex number makes with the positive real axis.
To find the magnitude of 625i, we can use the formula:
r = |a + bi| = sqrt(a^2 + b^2),
where a = 0 and b = 625:
r = |0 + 625i| = sqrt(0^2 + 625^2) = sqrt(390,625) = 625.
To find the angle θ, we can use the formula:
θ = arctan(b/a) (in the appropriate quadrant).
In this case, a = 0 and b = 625, so we have:
θ = arctan(625/0).
However, arctan(∞) is undefined, so we will need to consider the limit as a approaches 0 from the positive side:
θ = lim(a->0+) arctan(625/a) = π/2.
Therefore, the polar form of 625i is:
625i = 625(cos(π/2) + i*sin(π/2)).
Now, we can express x^4 = 625i in polar form:
x^4 = 625(cos(π/2) + i*sin(π/2)).
To find the fourth roots of 625(cos(π/2) + i*sin(π/2)), we need to take the fourth root of the magnitude and divide the angle by 4:
x = (625)^(1/4) * [cos(π/2 + 2kπ)/4 + i*sin(π/2 + 2kπ)/4],
where k is an integer ranging from 0 to 3.
Now we can calculate the solutions:
For k = 0: x = (625)^(1/4) * [cos(π/8) + i*sin(π/8)].
For k = 1: x = (625)^(1/4) * [cos(5π/8) + i*sin(5π/8)].
For k = 2: x = (625)^(1/4) * [cos(9π/8) + i*sin(9π/8)].
For k = 3: x = (625)^(1/4) * [cos(13π/8) + i*sin(13π/8)].
Graphically, we can represent the solutions as points on the complex plane. The solutions are the points where the fourth roots of 625i lie. Each solution corresponds to a value of k.
To plot these solutions, draw a set of axes where the x-axis represents the real part and the y-axis represents the imaginary part of the complex numbers. Then, locate the points (625)^(1/4) * [cos(π/8) + i*sin(π/8)], (625)^(1/4) * [cos(5π/8) + i*sin(5π/8)], (625)^(1/4) * [cos(9π/8) + i*sin(9π/8)], and (625)^(1/4) * [cos(13π/8) + i*sin(13π/8)] on the complex plane.