Use De Moivre's theorem to simplify the expression. Write the answer in a + bi form.
[2(cos 240° + i sin 240°)]^5
z = 2(cos 240° + i sin 240°) =
2 exp(4/3 pi i)
z^5 = 2^5 exp(20/3 pi i) =
32 exp(2/3 pi i) =
-16 + 16 sqrt(3) i
(2cis240°)^5 = 2^5 cis(5*240°)
= 32 cis1200°
= 32 cis120°
= -16 + 16√3 i
To simplify the expression using De Moivre's theorem, we need to raise the complex number to the power of 5.
De Moivre's theorem states that for any complex number z = r(cos θ + i sin θ) raised to the power of n, the result can be found using the following formula:
z^n = r^n(cos nθ + i sin nθ)
In this case, we have:
z = 2(cos 240° + i sin 240°)
n = 5
First, we need to calculate r, which represents the modulus or magnitude of the complex number. The modulus r can be found using the distance formula:
r = √(a^2 + b^2)
In this case, a = cos 240° and b = sin 240°.
a = cos 240° = -0.5
b = sin 240° = -√3/2
r = √((-0.5)^2 + (-√3/2)^2)
r = √(0.25 + 3/4)
r = √(1 + 3/4)
r = √(7/4)
r = √7/2
Now, we can apply De Moivre's theorem:
z^n = [2(cos 240° + i sin 240°)]^5 = (2)^5 (cos (5 * 240°) + i sin (5 * 240°))
= 32 (cos 1200° + i sin 1200°)
To simplify the expression, we can reduce 1200° to the equivalent angle within one full rotation (360°).
1200° = 360° * 3 + 120°
cos 1200° = cos 120°
sin 1200° = sin 120°
Therefore, the simplified expression is:
32 (cos 120° + i sin 120°)
In a + bi form, this can be written as:
32(cos 120°) + 32(i sin 120°)
= 32(-0.5) + 32(i * √3/2)
= -16 + 16i√3
So, the answer in a + bi form is -16 + 16i√3.