a. (2 - 2i√3)⁴ in polar form is 256(cos(-4π/3) + isin(-4π/3)) = 256cis(-4π/3)
b. (2 - 2i√3)⁴ in rectangular form is -128 + 128√3
To answer the question, we need to know what complex numbers are
What are complex numbers?
Complex numbers are numbers of the form z = x + iy
a. Complex numbers in polar form
Complex numbers in polar form z = r(cosθ + isinθ) where
- r = √(x² + y²) and
- θ = tan⁻¹(y/x)
Given that z = (2 - 2i√3)⁴ =
So,
So, converting to polar form
r = √(x² + y²)
= √[2² + (-2√3)²]
= √[4 + 4(3)]
= √[4 + 12]
= √16
= 4
θ = tan⁻¹(y/x)
θ = tan⁻¹(-2√3/2)
θ = tan⁻¹(-√3)
θ = -π/3
So, z = r(cosθ + isinθ)
= 4(cos(-π/3) + isin(-π/3))
Powers of complex numbers
A complex number z raised to power n is zⁿ = rⁿ(cosnθ + isin(nθ)]
z⁴ = (2 - 2i√3)⁴
= r⁴(cos4θ + isin4θ)
= 4⁴(cos(4 × -π/3) + isin(4 × -π/3))
= 256(cos(-4π/3) + isin(-4π/3))
= 256cis(-4π/3)
(2 - 2i√3)⁴ in polar form is 256(cos(-4π/3) + isin(-4π/3)) = 256cis(-4π/3)
b. Complex numbers in rectangular form
The complex number z = r(cosθ + isinθ) in rectangular form is z = x + iy where
Given that z⁴ = 256(cos(-4π/3) + isin(-4π/3)) in rectangular form,
x = rcosθ
= 256(cos(-4π/3)
= 256cos(-4 × 60°)
= 256cos(-240)
= 256cos(240)
= 256 × -1/2
= -128
y = rsinθ
= 256sin(-4π/3)
= 256sin(-4 × 60°)
= 256sin(-240)
= -256sin240
= -256 × -√3/2
= 128√3
So, z⁴ = x + iy
= -128 + 128√3
So, (2 - 2i√3)⁴ in rectangular form is -128 + 128√3
Learn more about complex numbers in polar form here:
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