Show that each complex nth root of a nonzero complex number w has the same magnitude.

Name: Problem 65, Chapter 8: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-07-07T17:56:41-08:00
Duration: 2 min 50 s
Channel: Foster Wisusik
Description: Show that each complex nth root of a nonzero complex number w has the same magnitude.

step 1 Okay, this question wants us to show that all nth roots of a complex number have the same magnit

Show that each complex nth root of a nonzero complex number...

Key Concepts

Complex Numbers and Modulus

Complex numbers consist of a real part and an imaginary part, and their magnitude (or modulus) is the distance from the origin in the complex plane. The modulus of a complex number z = a + bi is given by |z| = ?(a² + b²), which measures its 'size' regardless of direction.

Polar Form of Complex Numbers

Any nonzero complex number can be expressed in polar form as z = r (cos ? + i sin ?) or equivalently z = re^(i?), where r is the modulus and ? is the argument (angle). This form simplifies operations like multiplication, division, and finding roots by separating magnitude and directional components.

nth Roots of Complex Numbers

The nth roots of a complex number are the solutions to the equation z^n = w. In polar form, each nth root can be obtained by taking the nth root of the magnitude of w and dividing the argument by n, while adding integer multiples of 2?/n to capture all distinct roots. This process inherently shows that all the roots share the same magnitude.

Properties of Exponents in Complex Numbers

When raising a complex number in polar form to a power or extracting a root, the magnitude and argument behave independently under exponentiation. Specifically, if z^n = w, then taking the magnitude of both sides leads to |z|^n = |w|, and so every nth root of w has a magnitude equal to |w|^(1/n), irrespective of the root chosen.

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