Evaluate fourth root of -1

Problem

√(4,−1)

Solution

1. Express the number in polar form. The complex number −1 can be written as e(i*(π+2*k*π)) where k is an integer.

−1=cos(π)+i*sin(π)

2. Apply De Moivre's Theorem for roots. To find the nth roots of a complex number r*e(i*θ) use the formula r(1/n)*ei((θ+2*k*π)/n) for k=0,1,…,n−1

√(4,−1)=(−1)(1/4)=ei((π+2*k*π)/4)

3. Calculate the roots for k=0,1,2,3

k=0⇒e(iπ/4)=cos(π/4)+i*sin(π/4)=√(,2)/2+(i√(,2))/2

k=1⇒e(i(3*π)/4)=cos((3*π)/4)+i*sin((3*π)/4)=−√(,2)/2+(i√(,2))/2

k=2⇒e(i(5*π)/4)=cos((5*π)/4)+i*sin((5*π)/4)=−√(,2)/2−(i√(,2))/2

k=3⇒e(i(7*π)/4)=cos((7*π)/4)+i*sin((7*π)/4)=√(,2)/2−(i√(,2))/2

4. Combine the results into a single expression using ± notation.

√(4,−1)=±√(,2)/2±(i√(,2))/2

Final Answer

√(4,−1)=√(,2)/2±(i√(,2))/2,−√(,2)/2±(i√(,2))/2

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