Answer: Plain: the product is 24 ∠ 15° in polar form. This means that the magnitude of the product is 24 and the angle between the positive real axis and the line connecting the origin and the product is 15°, measured counterclockwise.
Step-by-step explanation: To multiply complex numbers in polar form, we multiply their magnitudes and add their angles. We can start by converting the given complex numbers from rectangular form to polar form:
6√5 - 6i = 6(√5 - i) = 12 ∠ -30°
(√3 + √3i) = √3(1 + i) = 2 ∠ 45°
where we have used the fact that ∠θ is the angle between the positive real axis and the line connecting the origin and the complex number a + bi, measured counterclockwise.
Now, we can multiply the two complex numbers in polar form:
(6√5 - 6i)(√3 + √3i) = 12 ∠ -30° * 2 ∠ 45°
= 24 ∠ 15°
Therefore, the product is 24 ∠ 15° in polar form. This means that the magnitude of the product is 24 and the angle between the positive real axis and the line connecting the origin and the product is 15°, measured counterclockwise.
To determine the quadrant of the complex plane in which the product lies, we note that the angle 15° is in the first quadrant (between 0° and 90°). Therefore, the product lies in the first quadrant of the complex plane.