Vereinfachen der Brüche: (1/x^k ) - (1+ x^-k ) / (1-x^k )

Question

Z.B:

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

Answer 1

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

Hauptnenner   x^k * ( 1-x^k ) also

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

= (  1-x^k   -  x^k -  x⁰)   ) /  ( x^k * ( 1-x^k ) )

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

= (  -2x^k   ) /  ( x^k * ( 1-x^k ) )          kürzen

=  -2    /   ( 1-x^k )

=  2    /   ( x^k - 1 )  

Answer 2

Bilde den Hauptnenner: x^k(1-x^k)

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

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

= (-2x^k)/(x^k(1-x^k)) --------<x^k kürzen

(-2)/((1-x^k)) 

=2/(x^k-1)