Expand Using De Moivre's Theorem 2+2i

Problem

2+2*i

Solution

1. Identify the complex number z=a+b*i where a=2 and b=2

2. Calculate the modulus r using the formula r=√(,a2+b2)

r=√(,2+2)

r=√(,8)

r=2√(,2)

3. Calculate the argument θ using tan(θ)=b/a Since the point (2,2) is in the first quadrant, θ=arctan(1)

θ=π/4

4. Apply De Moivre's Theorem to express the complex number in polar form z=r*(cos(θ)+i*sin(θ))

z=2√(,2)*(cos(π/4)+i*sin(π/4))

Final Answer

2+2*i=2√(,2)*(cos(π/4)+i*sin(π/4))

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