is there somebody to answer it correctly and mathematically.i would be thankful?

Question

factorize
a ^ (n) - b ^ (n) when n is odd and greater then 3.
It is to be expressed in simplest and general form of n

NO ..... IN BETWEEN PLS.

pls use a,b,n and the numbers only in the factorized qanswer.

Answer 1

In general, you can factor x^n -1 as the product of cyclotomic polynomials phi_k (x) where k ranges over the divisors of n. The degree of phi_k (x) is the number of integers <=k which are relatively prime to k. Call this number p(k). Then
a^n -b^n
will be the product of
b^p(k) * phi_k (a/b)
as k ranges over the divisors of n. This will be the prime factorization over the integers.

For example, the term when k=1 looks like (a-b). If n is prime, there is only one other factor (which is given by the other posters). If n is not odd, this other factor can be factored further.

Answer 2

a^n-b^n=a-b when n=1
=(a-b)(a^2+ab+b^2) when n=3
=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4) n=5
=(a-b)(a^6+a^5b+a^4b^2+a^3b^3+a^2b^4+ab^5+b^6) n=7
so by mathematical induction
a^n-b^n=(a-b)(a^(n-1)+a^(n-2)b+a^(n-3)b^2+..........
+ab^(n-2)+b^(n-1)

Answer 3

(a-b)(a^(n-1)b + a^(n-1)b^2 + ... + ab^(n-2) + b^(n-1))