Use DeMoivre's Theorem to find the indicated power of the following complex number
(-6 + 6i)^4
Please help, i don't get this guys therom at all...
DeMoivre's Theorem is a helpful tool when calculating powers of complex numbers. It can be stated as follows:
Let z = r(cos θ + i sin θ) be a complex number in trigonometric form, where r is the modulus and θ is the argument. For any positive integer n, DeMoivre's Theorem states that:
z^n = r^n [cos (nθ) + i sin (nθ)]
To use DeMoivre's Theorem to find the power of the complex number (-6 + 6i)^4, we need to first express the number in trigonometric or polar form.
Step 1: Finding the modulus (r)
The modulus (r) of a complex number can be calculated using the Pythagorean theorem:
r = √(a^2 + b^2)
In this case, a = -6 and b = 6, so the modulus is:
r = √((-6)^2 + 6^2)
= √(36 + 36)
= √(72)
= 6√2
Step 2: Finding the argument (θ)
The argument (θ) of a complex number can be found using the inverse tangent function:
θ = atan(b/a)
In this case, a = -6 and b = 6, so the argument is:
θ = atan(6/-6)
= atan(-1)
= -π/4
Step 3: Applying DeMoivre's Theorem
Now that we have the modulus (r = 6√2) and the argument (θ = -π/4), we can use DeMoivre's Theorem to calculate the power:
(-6 + 6i)^4 = (6√2)^4 [cos (4(-π/4)) + i sin (4(-π/4))]
Simplifying further:
(-6 + 6i)^4 = 1296 [cos (-π) + i sin (-π)]
Using the trigonometric identities cos(-π) = cos(π) = -1 and sin(-π) = sin(π) = 0, we get:
(-6 + 6i)^4 = 1296 (-1 + 0i)
= -1296
Therefore, the indicated power of the complex number (-6 + 6i)^4 is -1296.