demoivre theorem (2-2i)^8

Is there a question here? My crystal ball is on the blink.

yes

To find the value of (2-2i)^8 using De Moivre's Theorem, we can follow these steps:

Step 1: Convert the given complex number, 2-2i, into polar form.
To do this, we need to find the magnitude (r) and argument (θ) of the complex number. The magnitude (r) can be found using the Pythagorean theorem, which is the square root of the sum of the squares of the real and imaginary parts of the complex number.
r = √((2^2) + (-2^2)) = √(4 + 4) = √8 = 2√2

To find the argument (θ), we can use the inverse tangent function:
θ = arctan(-2/2) = arctan(-1) = -π/4

Therefore, the polar form of 2-2i is 2√2∠(-π/4).

Step 2: Apply De Moivre's Theorem.
De Moivre's Theorem states that for any complex number in polar form r∠θ, raised to the power of n, the result can be found by raising the magnitude to the power of n and multiplying the argument by n.

Applying De Moivre's Theorem, we have:
(2√2∠(-π/4))^8 = (2√2)^8∠(-π/4 * 8)
= 2^8 * (2√2)^0 ∠(-2π)

Simplifying further:
= 256 * 1 ∠(-2π)
= 256∠(-2π)

So, the value of (2-2i)^8 is 256∠(-2π), which is 256 when written in rectangular form.