using DeMoivre's theorem solve (-3,+3i)^4
To solve the expression (-3 + 3i)^4 using De Moivre's theorem, we can follow these steps:
1. Convert the given complex number into the trigonometric form:
-3 + 3i = r(cosθ + isinθ)
To find r and θ, we'll use the Pythagorean theorem and inverse tangent function:
r = √((-3)^2 + (3)^2) = √(9 + 9) = √18 = 3√2
tanθ = (Imaginary part) / (Real part) = 3 / -3 = -1
θ = arctan(-1) = -π/4 or 7π/4
Therefore, -3 + 3i can be expressed as 3√2 (cos(-π/4) + isin(-π/4)).
2. Apply De Moivre's theorem:
The formula for De Moivre's theorem states that (r(cosθ + isinθ))^n = r^n (cos(nθ) + isin(nθ)).
Substituting the values, we have:
(-3 + 3i)^4 = (3√2 (cos(-π/4) + isin(-π/4)))^4
Using De Moivre's theorem, we can simplify this as follows:
(-3 + 3i)^4 = (3√2)^4 (cos(4(-π/4)) + isin(4(-π/4)))
= 36 (-1 + 0i)
= -36
Therefore, (-3 + 3i)^4 simplifies to -36.