Simplify i^(1/3)

Problem

i^(1/3)

Solution

1. Express the imaginary unit i in polar form using Euler's formula e^(i*θ)=cos(θ)+i*sin(θ)

i=e^(i*(π/2+2*k*π))

2. Apply De Moivre's Theorem to find the cube roots by raising the expression to the power of 1/3

i^(1/3)=(e^(i*(π/2+2*k*π)))^(1/3)

3. Distribute the exponent to the angle for k=0,1,2

i^(1/3)=e^(i*(π/6+(2*k*π)/3))

4. Calculate the first root (k=0 in rectangular form.

e^(iπ/6)=cos(π/6)+i*sin(π/6)=√(,3)/2+1/2*i

5. Calculate the second root (k=1 in rectangular form.

e^(i(5*π)/6)=cos((5*π)/6)+i*sin((5*π)/6)=−√(,3)/2+1/2*i

6. Calculate the third root (k=2 in rectangular form.

e^(i(3*π)/2)=cos((3*π)/2)+i*sin((3*π)/2)=−i

Final Answer

i^(1/3)=√(,3)/2+1/2*i,−√(,3)/2+1/2*i,−i

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