Every nonzero complex number has exactly n distinct nth roots.
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The statement "every nonzero complex number has exactly n distinct nth roots" is true, and it is a fundamental concept in complex analysis.
To understand this, let's consider a nonzero complex number z. We can write z in polar form as z = re^(iθ), where r is the magnitude of z and θ is the argument of z.
To find the nth roots of z, we need to find complex numbers w that satisfy the equation w^n = z. Let's express w in polar form as w = se^(iϕ), where s is the magnitude of w and ϕ is the argument of w.
Substituting these forms into the equation, we get:
(se^(iϕ))^n = re^(iθ)
Using the properties of exponents, we can rewrite this equation as:
s^n * e^(inϕ) = re^(iθ)
For this equation to hold for all values of θ, we need to have:
s^n = r (1)
inϕ = θ (2)
Equation (1) tells us that the magnitude of w should be equal to the nth root of the magnitude of z. Since z is nonzero, it has a nonzero magnitude, and hence, the nth root of r exists and is nonzero.
Equation (2) determines the argument of w in terms of the argument of z. Since ϕ is an argument of w, nϕ is also an argument of w. This means that w can be rotated by an angle of 2π/n counter-clockwise n times around the origin to obtain n distinct roots.
Therefore, every nonzero complex number has exactly n distinct nth roots, as stated in the problem.
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