Question
Jan Villaroel
Topic: Algebra Posted 1 year ago
Every nonzero complex number has exactly n distinct nth roots.

Every nonzero complex number has exactly n distinct nth roots.

True or False?


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Koala Tutor
Aug 10, 2023
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True.

This statement is true. It can be proven by using De Moivre's Theorem, which states that for any complex number z = r(cosθ + isinθ), where r is the modulus of z and θ is the argument of z, the nth roots of z are given by:

z^1/n = r^(1/n)(cos[(θ + 2πk)/n] + isin[(θ + 2πk)/n]),

where k is an integer from 0 to n-1.

Since the argument θ can be expressed in multiple ways (e.g., θ + 2πk, θ + 4πk, etc.), there will be n distinct values for the nth roots. Therefore, every nonzero complex number has exactly n distinct nth roots.

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