Name Class Date Practice
Form K
Simplifying Rational Expressions
Simplify each expression. State any excluded values.
1.
3n - 15
12
2.
y+2
3. 2
y -4
12t 8
36t 6
15a - 50
4. 10a + 35
5.
q2 - 16
7q2 + 28q
6.
7.
m3 + 9m
6m2 - 3m
9z 2 - 36
8. 12z + 24
5x2 + x - 6
x2 - 1
9. The length of a rectangle is 8n + 24 and the width is 12n + 28. What is the
ratio of its length to its width? Simplify your answer.
10. The area of a rectangle is x2 + 6x - 16. Its width is x - 2. What is a simplified
expression for its length?
11. Writing Describe how you determine what values should be excluded when
simplifying a rational expression. Explain why this must be done.
12. Are the given factors opposites? Explain.
a. 5x - 2; 2 - 5x
b -t + 10; t + 10
c. 102 + 11d; -102 - 11d
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Name Class Date Practice (continued)
Form K
Simplifying Rational Expressions
13. A mother is packing away winter clothes into two rectangular tubs. Both hold
the same volume of clothes. The first tub has a length of 2b + 5, a width of
b - 3, and a height of 4b. The second tub has a width of 4b2 + 10b and a
length of b - 3. What is a simplified expression for the height of the second
tub? Show your work.
Simplify each expression. State any excluded values.
14.
x2 - 121
3x2 - 9x
v 3w 3
15. 2 3
v w
16.
5x2 - 41x + 42
x2 - 49
17.
2t 4 + t 3 - 28t 2
t 2 + 4t
18.
9m2 - 32m - 65
m2 - 25
19.
8a2 - 12a - 36
a2 - 9
x2 - 81
20. Writing Is x - 9 the same as x + 9? Explain.
21. Reasoning Is y = 4 an acceptable value for the expression
Explain.
3y 2 - 10y - 8
?
y 2 - 16
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Name Class Date Practice
Form K
Multiplying and Dividing Rational Expressions
Multiply.
1.
5n2
3n2
# n3
3a - 9
3. 3a - 6
5.
t
2. t - 3
# a a- 9
2
m2 - m - 20
m2 - 4m
z2
7. 2
z + 5z - 6
# m2m- 25
2
2
# 2z6z --7z15z+ 5
2
2
# tt ++ 12
# 54q4q- 18
4.
18q - 36
2q
6.
8v
6v 2 + 22v - 8
2
# 3v4v- 1
2
8. (3x 2 + 7x + 4)
9. Which of the following is the reciprocal of x2 - 2x - 63?
1
a.
b. (x + 7)(x - 9)
(x + 7)(x - 9)
# 9xx
2
3
- 4x
- 16x
1
c. x 9
Find the reciprocal of each expression.
10. x2 - 2x - 15
11.
6p2
7p2 - 12
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Name Class Date Practice (continued)
Form K
Multiplying and Dividing Rational Expressions
Divide.
6f - 6 6f - 6
12. 3f - 8 , f + 9
14.
18c - 27 2c - 3
,
9t 2 - 16 3t + 4
13.
12m - 20 3m - 5
27m , 9m
15.
2x2 - 23x + 56
x-8
, 5x
10x + 6
+3
Simplify each complex fraction.
16.
1
x-3
3
x-3
m
n +2
17. m
n +5
x
18. A shipping box has a base area of 4x 2 + 52x + 168 and a height of 4x + 28 .
What is the volume of the box?
1
19. Karl drives for (x2 - 100) hours at a rate of 5x 50 miles per hour. How far
does Karl drive?
20. Open-Ended Write two rational expressions whose product is 1.
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Name Class Date Practice
Form K
Adding and Subtracting Rational Expressions
Add or subtract.
8
19
2. 7p + 7p
2+2
1. n
n
3.
x-3 x+3
x - x
1 + b
4. 2 b b-2
4 - 2d
5. d
2
2k
7
6. 7 - 5k
Find the LCD of each pair of expressions.
2
7. 1
5, g
8.
2 , 7
3m2n mn2
2y y 2
10. 2
,
y +1 3
6
1
9. x - 3 , x +
4
11. Writing Explain how you can find the LCD when the GCF of the
denominators is 1.
12. What do you need to do to the numerators when using the LCD to add or
subtract the rational expressions? Explain.
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
Name Class Date Practice (continued)
Form K
Adding and Subtracting Rational Expressions
Add, subtract, and/or simplify.
2 +5
13. m
n
3
14. t + 3 + 5
x
15. 1 + y
a
2
16. b +
c(b - 2)
2.2 8.8
17. 2n - 3n
18.
3+2
w
10 - 7
19. What is the perimeter of a rectangular area rug that is
4p - 6
5 ft wide?
3+p
3 ft long and
20. Jennifer rode her bike to the store at a rate of 15 mi/h. She rode back home at a
rate of 10 mi/h. How far is it to the store if the round trip takes 1 hour?
21. Writing Why would you change a rational expression with a denominator
of x2 - 6x + 8 to (x - 2)(x - 4) when adding or subtracting rational
expressions?
22. Open-Ended Write a problem that uses addition of rational expressions in
which you need to find an LCD. Simplify the expression.
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Simplifying Rational
Expressions
1.
$
%&
3. ,
5.
6.
7.
8.
9.
1.
!"#
2.
4.
Multiplying & Dividing Rational
Expressions
'
-
2.
;) ≠ 0
3.
; 0 ≠ ±2
."/
'3"-4
/356
<"$
4.
; 7 ≠ −3.5
5.
; = ≠ 0, = ≠ −4,
6<
#@5A
; B ≠ ±1
@5D& 5ED
'(D"-)
'I"A
$
/!5A
6.
; H ≠ 0, H ≠
/
; J ≠ −2
(@"--)(@5--)
'@(@"')
; B ≠ 0, B ≠ 3
15. O; ,O ≠ 0, P ≠ 0
#@"A
@56
; B ≠ ±7
17.,) 2) − 7 ; ) ≠ 0, ) ≠ −4
18. ,
19.
ED"-'
; 0 ≠ ±5
D5#
$(/35')
35'
%(%5-)
(%"')(%5/)
3
(35')(3"/)
/<(<"/)
'<"/D(D5$)
(D"$)(D5#)
S(S5$)
I
'(I5A)
(@5-)(@"$)
10.
10. B + 8
11. Set each factor with a
variable in the denominator
equal to zero and solve for the
variable. The denominator
cannot equal zero.
12. a) yes b) no c) yes
13. ℎ = 2
16. ,
8.
1.
!
'@"$
9. a
'!56
14.
7.
#
; 7 ≠ ±3
20. No, the first expression
simplifies to the second but in
the first expression B ≠ 9
21. No, this would make the
denominator equal to 0.
11.
12.
13.
14.
15.
16.
17.
(@"#)(@5')
6T& "-/
AT&
U5E
'U"V
$
2.
$
!
/6
6T
3. −
4.
5.
6.
X"/
$"Y &
Y
-4Z & "$E
E
14.
'%"$
/@"6
15.,
/
'
D5/!
D5#!
@5-4
#
miles
'#Z
7. 5[
8. 3H / \/
9. (B − 3)(B + 4)
10. 3 0 / + 1
11. Multiply the denominators
by each other
12. Multiply the numerator by
the same thing you multiplied
the denominator by
'
-
A
@
"-5X
13.
18. B(B + 6)
19.
Adding & Subtracting Rational
Expressions
16.
/!5#D
D!
-V5#%
%5'
.5@
.
3] X"/ 5/X
X](X"/)
--
17. −
18.
19.
A!
#
'^
'$T"A
-#
ft
20. 6 mi
21. To make it easier to find the
LCD.