Final answer:
The common roots between Zⁿ-1=0 and Z^m-1=0 are the values of Z that make both equations equal to zero. The common roots can be found by solving each equation separately and then finding the values of Z that satisfy both equations.
Step-by-step explanation:
The common roots between Zⁿ-1=0 and Z^m-1=0 are the values of Z that make both equations equal to zero. Solving each equation separately, we can find the values of Z that satisfy each equation, and then find the common roots:
Zⁿ-1=0
To solve this equation, we can rewrite it as Zⁿ=1. The solutions to this equation are the n-th roots of unity. These roots are represented by:
Z = e(2πik)/n, where k = 0, 1, 2, ..., n-1.
Z^m-1=0
To solve this equation, we can rewrite it as Z^m=1. The solutions to this equation are the m-th roots of unity. These roots are represented by:
Z = e(2πik)/m, where k = 0, 1, 2, ..., m-1.
The common roots between the two equations are the values of Z that satisfy both equations. So, we need to find the values of k that make both expressions equal. This can be done by finding the values of k that make both (2πik)/n and (2πik)/m integers. By finding the least common multiple (LCM) of n and m and setting (2πik)/n = (2πik)/m = 2πik/LCM(n,m), we can find the common roots.
Therefore, the common roots between Zⁿ-1=0 and Z^m-1=0 are given by:
Z = e(2πik)/LCM(n,m), where k = 0, 1, 2, ..., LCM(n,m)-1.