Question

Find the indicated power using De Moivre's theorem. Express your fully simplified answer in the form a + bi.

A) (2i^20)
B) (2^20i)
C) (2^20)
D) (2^10i)

Answer 1

To find the indicated power using De Moivre's theorem, express the number in trigonometric form, raise it to the desired power, and convert it back to rectangular form.

Step-by-step explanation:

To find the indicated power using De Moivre's theorem, we need to express the number in trigonometric form, raise it to the desired power, and then convert it back to rectangular form. In this case, we have (2i^20).

First, let's express 2i in trigonometric form. We have r = sqrt(2^2 + 0^2) = 2 and theta = arctan(0/2) = 0.

Now, we can raise 2i to the power of 20. Using De Moivre's theorem, we have (r(cos(theta) + isin(theta)))^20 = 2^20(cos(0*20) + isin(0*20)). Since cos(0) = 1 and sin(0) = 0, the result is 2^20(1 + 0i) = 2^20.

Therefore, the fully simplified answer is C) (2^20).