Expand Using De Moivre's Theorem (1+i)^20

Problem

(1+i)20

Solution

1. Identify the complex number z=1+i and determine its modulus r

r=√(,1+1)

r=√(,2)

2. Determine the argument θ of the complex number.

tan(θ)=1/1

θ=π/4

3. Write the complex number in polar form.

z=√(,2)*(cos(π/4)+i*sin(π/4))

4. Apply De Moivre's Theorem, which states zn=rn*(cos(n*θ)+i*sin(n*θ))

(1+i)20=(√(,2))20*(cos(20⋅π/4)+i*sin(20⋅π/4))

5. Simplify the exponent and the arguments.

(√(,2))20=(2(1/2))20=2=1024

20⋅π/4=5*π

6. Evaluate the trigonometric functions at the simplified argument.

cos(5*π)=−1

sin(5*π)=0

7. Calculate the final result by multiplying the modulus by the evaluated trigonometric terms.

1024*(−1+i(0))=−1024

Final Answer

(1+i)20=−1024

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