Textbook Question
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 + i)⁵
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
8:37 minutesProblem 5.2.65
Textbook Question
In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)
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Identify the given complex number in polar form: \(9(\cos 30^\circ + i \sin 30^\circ)\). Here, the modulus \(r = 9\) and the argument \(\theta = 30^\circ\).
Recall that to find the complex square roots of a number in polar form \(r(\cos \theta + i \sin \theta)\), we use the formula for the \(n\)th roots: \(\sqrt[n]{r} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)\), where \(k = 0, 1, ..., n-1\). Since we want square roots, \(n=2\).
Calculate the modulus of the roots by taking the square root of \(r\): \(\sqrt{9} = 3\).
Calculate the arguments of the roots by dividing the original argument plus \(360^\circ k\) by 2 for \(k=0\) and \(k=1\): \(\frac{30^\circ + 360^\circ \times 0}{2}\) and \(\frac{30^\circ + 360^\circ \times 1}{2}\).
Write the two roots in polar form as \(3 \left( \cos \alpha + i \sin \alpha \right)\) where \(\alpha\) are the two arguments found in the previous step.
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8mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers in Polar Form
Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form simplifies multiplication, division, and finding roots by working with magnitudes and angles separately.
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Complex Numbers In Polar Form
De Moivre's Theorem
De Moivre's theorem states that for a complex number in polar form, raising it to the power n results in r^n (cos nθ + i sin nθ). Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, adding multiples of 360°/n for all roots.
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Finding Complex Roots
To find the complex nth roots of a number, calculate the nth root of the magnitude and determine the arguments by dividing the original angle by n and adding k(360°/n) for k = 0, 1, ..., n-1. This yields all distinct roots evenly spaced around the circle.
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