Simplify square root of 3i

Problem

√(,3*i)

Solution

1. Express the complex number in polar form. The number 3*i has a magnitude r=3 and an angle θ=π/2

3*i=3*e^(iπ/2)

2. Apply the square root property for complex numbers in polar form, which is √(,r*e^(i*θ))=√(,r)*e^(iθ/2)

√(,3*i)=√(,3)*e^(iπ/4)

3. Convert the result back to rectangular form using Euler's formula e^(i*x)=cos(x)+i*sin(x)

√(,3)*e^(iπ/4)=√(,3)*(cos(π/4)+i*sin(π/4))

4. Substitute the known values for cos(π/4)=√(,2)/2 and sin(π/4)=√(,2)/2

√(,3)*(√(,2)/2+i√(,2)/2)

5. Distribute the √(,3) to find the final simplified expression.

√(,6)/2+√(,6)/2*i

Final Answer

√(,3*i)=√(,6)/2+√(,6)/2*i

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