Find each root. See Example 3. -√((1)/(32))

Name: Problem 48, Chapter 0: Trigonometry
Uploaded: 2023-02-20T05:12:23-08:00
Duration: 1 min 19 s
Channel: Mukesh Devi
Description: Find each root. See Example 3. -√((1)/(32))

Key Concepts

Nth Root Extraction and Periodicity

Finding the nth roots of a complex number involves computing the nth root of its magnitude and determining the n distinct angles by adding multiples of 2?. The general formula is given by r^(1/n) (cos((?+2?k)/n) + i sin((?+2?k)/n)) for k = 0, 1, ..., n-1. This procedure ensures that all distinct roots are found, thanks to the periodicity of the trigonometric functions with a period of 2?.

De Moivre's Theorem

De Moivre's Theorem provides a formula for raising complex numbers in polar form to integer powers and extracting nth roots. Specifically, if a complex number is given as r (cos? + i sin?), then its nth roots are found by taking the nth root of the modulus, r^(1/n), and dividing the angle by n, adding multiples of 2? to generate all distinct roots. This theorem is key to solving equations involving complex exponentiation.

Complex Numbers

Complex numbers extend the real numbers by including an imaginary unit, denoted by i, where i² = -1. They can be expressed in the form a + bi, where a is the real part and b is the imaginary part. This concept is crucial when finding roots of numbers that may not be real, as every non-zero complex number has exactly n distinct nth roots in the complex plane.

Polar Representation

The polar form of a complex number expresses it in terms of its magnitude (or modulus) and angle (or argument). A complex number z can be written as r (cos? + i sin?) or r e^(i?), where r is the distance from the origin and ? is the angle measured from the positive real axis. This representation simplifies the process of root extraction since the magnitude and angle can be manipulated separately.

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