how does k(x)^k-1 simplify to x^k ?

Question

I'm doing some mathematical inductions and I come down to (k+1)x^k and (k+1)(k)(x)^k-1 so i'm wondering how k(x)^k-1 simplifies to x^k ?

Answer 1

It doesn't simplify to x^k
If it did then the following equation would be true for all k and x
k * x^k - 1 = x^k

Take an example value for x. when x = 1, the left side equals k - 1.
but the right side equals 1. These are obviously not equal for all k
Therfore these statements aren't equal.

Answer 2

k(x^(k - 1)) = x^k
k = (x^k)/(x^(k - 1))
k = x^(k - (k - 1))
k = x^(k - k + 1)
k = x^(1)
k = x^1
k = x

ANS : k = x