The given equation is x^2021 * x^2020 * x^2019 = 1. We need to find the possible roots of this equation.




To simplify the equation, we can combine the exponents of x:
x^(2021 + 2020 + 2019) = 1
x^6060 = 1


To find the roots of unity, we need to solve the equation x^6060 = 1.


De Moivre's theorem states that for any complex number z = r(cosθ + isinθ), where r is the modulus (or absolute value) of z and θ is the argument (or angle) of z, the nth power of z can be expressed as z^n = r^n(cos(nθ) + isin(nθ)).

In this case, z = 1, so we have:
1^n = 1(cos(0) + isin(0))

For the equation x^6060 = 1, we have:
x^6060 = 1(cos(0) + isin(0))

Comparing the two equations, we can equate the exponents:
6060n = 0 + 2πk, where k is an integer


To find the possible values of n, we divide both sides of the equation by 6060:
n = 0/6060 + 2πk/6060
n = 0 + πk/3030


Substituting the values of n back into the equation x^6060 = 1, we get:
x^(πk/3030) = 1

Taking the πth root of both sides, we have:
x^(k/3030) = 1^(1/π)
x^(k/3030) = 1

Since any non-zero number raised to the power of zero is 1, we can conclude that the possible values of x are any non-zero complex numbers.

Therefore, the equation x^2021 * x^2020 * x^2019 = 1 has an infinite number of possible roots, which are all non-zero complex numbers.


The equation x^2021 * x^2020 * x^2019 = 1 has an infinite number of possible roots, which are all non-zero complex numbers.