I'm trying to use DeMoirre's Theorem to compute and express this in rectangular form. Here's the expression:
(-1 + i)^5
The ^ means it's to that power. It's to the fifth power.
Any help would be appreciated.
Figured it out myself. 4 - 4i is the answer.
Great job on figuring out the answer! However, I'd be happy to provide an explanation of how to use De Moivre's Theorem to find the answer to your problem.
De Moivre's Theorem is a formula that allows us to raise complex numbers to a power. It states that for any complex number z in polar form (r, θ), where r is the magnitude and θ is the angle, and any non-negative integer n:
(z^n) = (r^n, nθ)
To express the complex number (-1 + i) in polar form, we need to find its magnitude and angle. The magnitude can be calculated using the Pythagorean theorem:
|r| = sqrt((-1)^2 + 1^2) = sqrt(2)
To find the angle, we can use the inverse tangent function:
θ = atan(1/-1) = atan(-1) = -π/4 (since arctan(-1) = -π/4)
Now that we have the polar form for (-1 + i), which is (√2, -π/4), we can apply De Moivre's Theorem to raise it to the power of 5:
((-1 + i)^5) = (√2^5, 5(-π/4))
Now, let's simplify this expression:
√2^5 = 2^(5/2) = 2 * √2(√2^2) = 2 * √2(2)
Since the angle -π/4 is a multiple of 2π, it wraps around to 0:
5(-π/4) = -5(π/4) = -5π/4 + 2π = -5π/4 + 8π/4 = 3π/4
Therefore, we have:
((-1 + i)^5) = (2 * √2(2), 3π/4)
Now we can convert this back to rectangular form:
To convert a complex number from polar form (r, θ) to rectangular form, we use the following equations:
x = r * cos(θ)
y = r * sin(θ)
Using these equations, we can find the rectangular form of the complex number:
x = (2 * √2(2)) * cos(3π/4) = 2 * √2(2) * (-√2/2) = -2
y = (2 * √2(2)) * sin(3π/4) = 2 * √2(2) * (√2/2) = 2√2
Hence, the rectangular form of the complex number (-1 + i)^5 is -2 + 2√2i.