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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
Koala Ai Support
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.