Verified answer

Question A)

... ( (a - b^(-1) )^(-1) ) / (a b^(-1) )^(-1)

or (a b^(-1) ) / ( (a - b^(-1) )

or (a/b) / ( a - 1/b )

or a / ( ab - 1 )

Question B)

... (81 x^(6) y^(12) )^(3/4)

= 81^(3/4) x^(6 * 3/4) y^(12 * 3/4)

= 3^(4*3/4) x^(18/4) y^(36/4)

= 3^(3) x^(9/2) y^(9)

= 27 x^(9/2) y^(9)

Question C)

.. ( 3 √( 27^(6) y^(14) ) )^2 / √( 9 x^(2) y^(6) )^(3) )^(-1)

= 3^(2) 27^(6) y^(14) ( 9 x^(2) y^(6) )^(3/2)

= 3^(2) 3^(3*6) 3^(2 * 3/2) x^(2 * 3/2) y^(14) y^(6 * 3/2)

= 3^(2 + 18 + 3) x^(3) y^(14 + 9)

= 3^(23) x^(3) y^(23)