Textbook Question
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 20° + i sin 20°)]³
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
6:56 minutesProblem 29
Textbook Question
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (−2 − 2i)⁵
Verified step by step guidance1
Express the complex number \((-2 - 2i)\) in polar form. First, find the modulus \(r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4}\).
Next, find the argument \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right) = \tan^{-1}\left(\frac{-2}{-2}\right)\). Adjust \(\theta\) to the correct quadrant since both parts are negative.
Apply DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)-th power is \(r^n (\cos(n\theta) + i \sin(n\theta))\). Here, \(n = 5\).
Calculate \(r^5\) and multiply the argument by 5 to get \(5\theta\). Then write the expression \(r^5 (\cos(5\theta) + i \sin(5\theta))\).
Convert the result back to rectangular form by evaluating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) for the imaginary part.
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6mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
DeMoivre's Theorem
DeMoivre's Theorem states that for a complex number expressed in polar form as r(cos θ + i sin θ), its nth power is given by r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.
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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
Conversion Between Rectangular and Polar Forms
To apply DeMoivre's Theorem, a complex number must be converted from rectangular form (a + bi) to polar form (r, θ), where r is the magnitude √(a² + b²) and θ is the argument arctan(b/a). This conversion is essential for using trigonometric identities in complex number powers.
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Rectangular Form of Complex Numbers
After applying DeMoivre's Theorem in polar form, the result is converted back to rectangular form a + bi by calculating a = r^n cos(nθ) and b = r^n sin(nθ). Writing the answer in rectangular form makes it easier to interpret and use in further calculations.
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