find the cube roots of -216
answer in polar form and complex
To find the cube roots of -216, we can start by expressing -216 in its polar form.
Polar form of a complex number is expressed as:
z = r * (cosθ + i * sinθ)
where r represents the magnitude of the complex number and θ represents the angle it makes with the positive real axis.
To find the polar form of -216, we need to calculate the magnitude and the angle of -216.
Magnitude (r):
The magnitude of a complex number can be calculated using the absolute value:
|z| = √(Re² + Im²)
In this case, as -216 is a real number, the imaginary part is zero. Therefore, the magnitude |z| will be:
|r| = √((-216)² + 0²)
|r| = √(46656)
|r| = 216
Angle (θ):
The angle θ can be found using the formula:
θ = arctan(Im / Re)
As the imaginary part of -216 is zero, the angle will be:
θ = arctan(0 / -216)
θ = arctan(0)
θ = 0
Therefore, the polar form of -216 is:
z = 216 * (cos0 + i * sin0)
Now, we can find the cube roots by taking the cube roots of the magnitude and dividing the angle by 3:
Cube root 1:
r₁ = ∛|z| = ∛216
Angle₁ = θ / 3 = 0 / 3 = 0
Cube root 2:
r₂ = ∛|z| = ∛216
Angle₂ = (θ + 2π) / 3 = (0 + 2π) / 3 = 2π / 3
Cube root 3:
r₃ = ∛|z| = ∛216
Angle₃ = (θ + 4π) / 3 = (0 + 4π) / 3 = 4π / 3
Finally, let's express the cube roots in polar form:
Cube root 1: ∛216 * (cos0 + i * sin0)
Cube root 2: ∛216 * (cos(2π / 3) + i * sin(2π / 3))
Cube root 3: ∛216 * (cos(4π / 3) + i * sin(4π / 3))
Simplifying the angles and evaluating the cube roots will give the exact complex numbers in polar form.
-216 = 216 e^(i*pi) = 6^3*e^(i*pi) =
6^3*e^(3 pi)= 6^3*e^(5 pi)
The cube roots of those number are is
6 e^(i*pi/3), 6 e^(i*pi) and
6 e(i*5pi/3)
The second of those numbers is -6. You can put the other two roots in complex form by using the identity
e^ix = cos x + i sin x