Download Dividing Signed Numbers

9.5
Dividing Signed Numbers
9.5
OBJECTIVES
1. Find the quotient of two signed numbers
2. Recognize that division by zero is undefined
You know from your work in arithmetic that multiplication and division are related operations. We can use that fact, and our work of the last section, to determine rules for the
division of signed numbers. Every division problem can be stated as an equivalent multiplication problem. For instance,
15
3
5
because
15 5 3
24
4
6
because
24 (6)(4)
30
6
5
because
30 (5)(6)
The examples above illustrate that because the two operations are related, the rule of
signs that we stated in the last section for multiplication is also true for division.
Rules and Properties:
Dividing Signed Numbers
The quotient of two numbers with different signs is negative.
The quotient of two numbers with the same sign is positive.
Again, the rule is easy to use. To divide two signed numbers, divide their absolute
values. Then attach the proper sign according to the rule above.
Example 1
Dividing Signed Numbers
Divide.
Positive
(a)
Positive
Positive
Negative
(b)
Negative
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Negative
(c)
Positive
Positive
(d)
Negative
Positive
(e)
Negative
36
9
4
Positive
42
6
7
Negative
75
25
3
15.2
4
3.8
Negative
Negative
703
704
CHAPTER 9
THE REAL NUMBER SYSTEM
DIVIDING SIGNED NUMBERS
SECTION 9.5
705
CHECK YOURSELF 2
Divide if possible.
0
3
(a)
(b)
5
0
(c)
7
0
(d)
0
9
Recall that the fraction bar serves as a grouping symbol. This means that all operations
in the numerator and denominator should be performed separately. Then the division is
done as the last step. Example 3 illustrates this property.
Example 3
Dividing Signed Numbers
Evaluate each expression.
(a)
(6)(7)
42
14
3
3
Multiply in the numerator, then
divide.
(b)
3 (12)
9
3
3
3
Add in the numerator, then
divide.
(c)
4 (2)(6)
4 (12)
6 2
6 2
Multiply in the numerator. Then
add in the numerator and
subtract in the denominator.
16
2
8
Divide as the last step.
CHECK YOURSELF 3
Evaluate each expression.
(a)
4 (8)
6
(b)
3 (2)(6)
5
(c)
(2)(4) (6)(5)
(4)(11)
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In our last example in this section, we will evaluate expressions containing several operations. Recall the order of operations:
Rules and Properties:
NOTE
Order of Operations
Please
1. Perform all operations inside of grouping symbols. Grouping symbols include
parentheses, brackets, absolute value signs, and fraction bars.
Excuse
2. Apply all exponents (powers).
My Dear
Aunt Sally
3. Do all multiplication and division, from left to right.
4. Do all addition and subtraction, from left to right.
706
CHAPTER 9
THE REAL NUMBER SYSTEM
Example 4
Evaluating Expressions
Evaluate each expression.
(a) 4 2 (5 7)2
bars are grouping symbols.
Evaluate the numerator and
denominator separately.
NOTE In algebra, we rarely
change an improper fraction to
a mixed number.
(b)
Evaluate inside parentheses first
424
Apply the exponent
48
Multiply
12
Add
3 (2)3
7 3
3 (8)
4
38
4
11
4
11
4
CHECK YOURSELF 4
Evaluate each expression.
(a) 35 (3 7)3
(b)
2 3 (1 5)2
(3)3 (2)4
CHECK YOURSELF ANSWERS
1. (a) 5; (b) 4; (c) 6; (d) 12; (e) 5
2. (a) 0; (b) undefined; (c) undefined; (d) 0
1
46
3. (a) 2; (b) 3; (c)
4. (a) 29; (b)
2
11
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NOTE Remember that fraction
4 2 (2)2
Name
9.5
Exercises
Section
Date
Divide.
1.
4.
20
4
2.
24
8
5.
70
14
50
5
3.
6.
48
6
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
96
8
13.
14.
15.
16.
0
8
17.
32
8
52
7.
4
56
8.
7
75
9.
3
60
10.
15
0
11.
8
125
12.
25
13.
16.
19.
9
1
14.
20
2
17.
17
1
150
22.
6
20.
10
0
15.
18
0
18.
27
1
29.4
23.
4.9
21.
144
16
25.9
24.
3.7
25.
8
32
26.
6
30
27.
24
16
28.
25
10
29.
28
42
30.
125
75
Perform the indicated operations.
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31.
34.
(6)(3)
2
(7)(8)
14
32.
35.
(9)(5)
3
24
4 8
33.
36.
ANSWERS
(8)(2)
4
36
7 3
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
37.
12 12
3
11 7
40.
14 8
38.
14 4
6
75
41.
22
39.
55 19
12 6
10 6
42.
44
41.
42.
707
ANSWERS
43.
43.
15 (3)
3 (1)
44.
21 (4)
3 2
45.
(9)(6) 10
18 (4)
46.
4 2(6)
14 (6)
47.
(3)(6) (4)(8)
6 (4)
48.
(5)(2) (4)(5)
4 2
49.
2(5) 4(6 8)
3(4 2)
50.
(3)(5) 3(5 8)
4(8 6)
44.
45.
46.
47.
48.
49.
50.
51.
52.
Evaluate the following. Remember to use the order of operations.
53.
51. 5 3 (4 6)2
52. 8 2 (3 6)3
53. 20 2 10 2
54. 6 3 2 9
54.
55.
56.
55.
57.
58.
4 (3)2
7 4 3
57. 60 (3) (4) 23 4
56.
4 (2)3
52 (2)2
58.
16 (2) (4)
23 4
60.
4 (3) 1
2 2 (2)
59.
59.
60.
7 (1)
32 9 (3)
Answers
5. 10
3. 8
19. 17
17. Undefined
2
29.
3
43. 3
57. 76
708
31. 9
45. 2
59.
2
3
33. 4
47. 5
7. 13
9. 25
11. 0
21. 9
23. 6
35. 2
37. 8
1
25. 4
39. 2
51. 17
53. 10
49. 3
13. 9
15. 12
3
27. 2
41. Undefined
1
55.
5
© 2001 McGraw-Hill Companies
1. 5
Using Your Calculator
to Multiply and Divide
Signed Numbers
Finding the product of two signed numbers using a calculator is relatively straightforward.
Example 1
Multiplying Signed Numbers
Find the product. 457 (734)
457 734 335438
CHECK YOURSELF 1
Find the products.
(a) 36 (91)
(b) 12 (284)
Finding the quotient of signed numbers is also straightforward.
Example 2
Dividing Signed Numbers
Find the quotient. 376 16
376 16 23.5
CHECK YOURSELF 2
Find the quotient. 7865 242
We can also use the calculator to raise a signed number to a power.
Example 3
Raising a Number to a Power
Evaluate.
(3)6
© 2001 McGraw-Hill Companies
x
3 y 6 729
CHECK YOURSELF 3
Evaluate.
(2)9
CHECK YOURSELF ANSWERS
1. (a) 3276; (b) 3408
2. 32.5
3. 512
709
Name
Section
Calculator Exercises
Date
Use your calculator to multiply and divide the following. Where it’s appropriate, round to
the nearest hundredth.
ANSWERS
1.
1. 25 (21)
2. 15 (45)
3. 78 (12)
4. (56) 31
5. (34) (28)
6. (71) (19)
7. 345 (25)
8. 128 (28)
9. (564) 36
10. (232) 52
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
11. (456) (28)
12. (128) (42)
13. (28) (14)
14. (456) (124)
15. (4)5
16. (5)4
13.
14.
15.
16.
Answers
710
3. 936
15. 1024
5. 952
7. 13.8
9. 15.67
11. 16.29
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1. 525
13. 2