[division of rational expressions][section 54]

(1.)   4a - 16
_______ ÷ (3a - 12)         here is the problem
3a

=     4a - 16           1            multiply by the reciprocal
__________  *  __________
3a           3a - 12

=    4(a - 4)         1
________  *  ___________
3a         3(a - 4)             factor

=    4
____           cancel and multiply
9a

(2.)  6z - 27
_______ ÷ (4z - 18)              here is the problem
4z

=    3(2z - 9)
____________  ÷  2(2z - 9)       factor
4z

=    3(2z - 9)           1
____________ *  __________     multiply by the reciprocal
4z           2(2z - 9)

=     3
_____           cancel and multiply
8z

(3.)   (2x)3          16x2
_________ ÷  _______          here is the problem
(4xy)3         8y2z3

=      (2x)3           8y2z3
__________  *  __________ multiply by the reciprocal
(4xy)3           16x2

=    8x3            8y2z3
_________  *  __________   cube 2 & 4 & multiply exponents
64x3y3           16x2

64x3y2z3

1024x5y3

=         z3
______________         reduce, subtract exponents
16x2y

(4.)    a2 - b2           a + b
________  ÷   __________
x2 - y2           x - y         here is the problem

=   (a - b)(a + b)         (x - y)    factor and multiply
__________________  *  _________  by the reciprocal
(x - y)(x + y)        (a + b)

=   a - b
_____
x + y         cancel and multiply

(5.) x - y       5x2 - 5y2
_____  ÷   __________             here is the problem
x + y       3x - 3y

=    (x - y)         5(x2 - y2)
___________  ÷   ____________
(x + y)          3(x - y)        factor

=  (x - y)          5(x - y)(x + y)
_________  ÷   _________________
(x + y)            3(x - y)       factor

=    (x - y)             3(x - y)         multiply by the
___________  *   _______________    reciprocal
(x + y)         5(x - y)(x + y)

=    3(x - y)
___________
5(x + y)       cancel and multiply

(6.)   x2 + 2x + 1
______________  ÷   (x + 1)   here is the problem
3x

=   (x + 1)2            1           factor and multiply
___________  *  __________  by the reciprocal
3x          (x + y)

=    (x + 1)
___________
3x                cancel and multiply

(7.)   a3 - 6a2 + 8a        2a - 4
______________  ÷  ____________      here is the problem
5a           10a - 40

=    a(a - 4)(a - 2)        10(a - 4)   factor and multiply
___________________  *  ____________  by the reciprocal
5a               2(a - 2)

=     (a - 4)2       cancel and multiply

(8.)      k                       1
_______________   ÷   ___________
k2 - 6k + 9            k - 3

=        k                 (k - 3)    factor and multiply
_________________  *  ____________ by the reciprocal
(k - 3)2
1

=   k/(k - 3)             cancel and multiply

(9.)   a2 - 49           3a - 21
___________  ÷   _____________    here is the problem
(a + 7)2           2a + 14

=   (a - 7)(a + 7)           2(a + 7)   factor and multiply
__________________  *  _____________ by the reciprocal
(a + 7)2            3(a - 7)

=   2/3            cancel and multiply

(10.)               b2 + 8b + 15
(b2 - 9) ÷   ______________    here is the problem
2b + 10

=   (b - 3)(b + 3)            2(b + 5)      factor and multiply
_________________  *   ________________ by the reciprocal
1                (b + 5)(b + 3)

=   2(b - 3)           cancel and multiply