Final answer:
The correct square roots of 9i in rectangular form are 3 + 3i and -3 + 3i. This is found using polar representation and De Moivre's Theorem to determine the magnitude and angles of the square roots.
Step-by-step explanation:
To find the square roots of 9i in rectangular form, we should express the complex number 9i in its polar form first. The magnitude of 9i is 9, and its angle θ with the positive x-axis (in radians) can be determined by the fact that 9i lies on the positive imaginary axis, which corresponds to an angle of π/2 or 3π/2. Using De Moivre's Theorem, the nth roots of a complex number can be found by dividing the angle θ by n and finding the magnitude by taking the nth root of the magnitude of the original complex number.
In this case, to find the square roots, we divide the angles π/2 and 3π/2 by 2 to get π/4 and 3π/4, and the square root of the magnitude 9 is 3. Hence, the two square roots of 9i are 3(cosπ/4 + isinπ/4) and 3(cos3π/4 + isin3π/4).
These can be simplified to:
- 3(cosπ/4 + isinπ/4) = 3 + 3i
- 3(cos3π/4 + isin3π/4) = -3 + 3i
So the correct square roots of 9i in rectangular form are 3 + 3i and -3 + 3i.