Use DeMoivre's Theorem to find the indicated power of the following complex number.

Question

Quizplus · Accepted Answer

First, we need to convert the given complex number from polar form to rectangular form. Let z be the given complex number. Then, we have:

z = 11eb80b1_af8c_0d63_a291_c16dbe4d5e1e_TB8633_11
= 11(cos(321°) + i sin(321°))    (using Euler's formula)
= 11(-0.766 + 0.643i)             (using trigonometric ratios)

Now, we can use DeMoivre's Theorem to find z^n for any positive integer n. DeMoivre's Theorem states that:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

Therefore, we have:

z^2 = (11(cos(321°) + i sin(321°)))^2
= 11^2(cos(2*321°) + i sin(2*321°)) 
= 11^2(cos(642°) + i sin(642°))
= 11^2(cos(282°) + i sin(282°))    (since cos(x) = cos(360-x) and sin(x) = -sin(360-x) when x is in degrees)

Using the same formula, we can find z^4:

z^4 = (11(cos(321°) + i sin(321°)))^4
= 11^4(cos(4*321°) + i sin(4*321°)) 
= 11^4(cos(1284°) + i sin(1284°))
= 11^4(cos(204°) + i sin(204°))    (since cos(x) = cos(360-x) and sin(x) = -sin(360-x) when x is in degrees)

Finally, we can find |z^4| (the magnitude or absolute value of z^4) as follows:

|z^4| = |11^4(cos(204°) + i sin(204°))|
= 11^4 |cos(204°) + i sin(204°)|
= 11^4

Therefore, the indicated power of the given complex number is 4, and the correct answer is B) 64.

Use DeMoivre's Theorem to Find the Indicated Power of the Following