Question

Find all of square roots of 36i and write the answers in rectangular (standard) form

Answer 1

To find the square roots of 36i, we first need to write 36i in polar form:

r = √(0^2 + 36^2) = 36
θ = tan^-1(0/36) + kπ = kπ (where k is an integer)

Since 36i lies on the positive imaginary axis, we can choose k = 1/2 so that θ = π/2.

Thus, 36i in polar form is 36(cos(π/2) + i sin(π/2)).

Now we can find the square roots of 36i by taking the square root of the magnitude and dividing the angle by 2:

√36(cos(π/4) + i sin(π/4)) = ± 3(cos(π/8) + i sin(π/8))

√36(cos(9π/4) + i sin(9π/4)) = ± 3(cos(9π/8) + i sin(9π/8))

Therefore, the square roots of 36i in rectangular form are:

±3(cos(π/8) + i sin(π/8)) and ±3(cos(9π/8) + i sin(9π/8))