Solve basic Complex number problems Find squareroot i Find i

Solve basic Complex number problems

Find squareroot i. Find i^i. Find squareroot 1 + squareroot 3i. Find all the sixth roots of unity. Find all the solutions of the given equations: z^4 = -1 z^3 = -8 z^3 = -8. z^3 = -125i

Solution

10) a) i

e^ib = cosb + i sinb

Now, e^i(/2) = cos(/2) + i sin(/2)

==> e^i(/2) = 0 + i(1) = i

==> i = (e^i(/2))

==> [e^i(/2)]^1/2

==> e^i(/4)

Hence i = e^i(/4)

b) iⁱ = [e^i(/2)]ⁱ

==> e^{i^2 (/2)}

==> e^-(/2)     since i² = -1

Hence iⁱ = e^-(/2)

c) (1 + 3i)

multiply and divide by 2 inside sqaure root

==> (1 + 3i) = [2 (1 + 3i)/2]

==> 2 [1/2 + 3i/2]

==> 2 e^i/3            since e^i/3 = cos(/3) + isin(/3) = 1/2 + 3i/2

==> 2 (e^i/3 )^1/2

==> 2 e^i/6

Hence (1 + 3i) = 2 e^i/6