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1.02 Order of Operations, Signed Numbers, Absolute Value from Intermediate Algebra: One Step at a Time © 1998 Pages 14-26 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
In the first section, different number systems were defined, and properties relating to addition (subtraction) and multiplication (division) of numbers within those systems were identified. In order to execute these basic operations it is necessary to establish a few traditions (agreements!) as to how these operations will be carried out. The “signs” of the numbers must also be addressed. In this section it is preferred that you use the calculator “in your head.” In the next section, you will be encouraged to use the calculator “in your hand.” If you do not already have one, be looking for a good sale on inexpensive calculators! ORDER
OF OPERATIONS In your head, perform the following calculations: 1.
4 + 6 ·
2 = _______
2.
16 ÷ 8 ·
2 = _______ 3.
5 ·
2 2 = _______
4.
(3 + 4) 2 = _______ Solutions: 1. If addition is performed first, the answer is 20. If multiplication is performed first, the answer is 16.
2. If division is performed first, the answer is 4. If
multiplication is performed first, the answer is 1. 3. If multiplication is performed first, the answer is 100. If the power is performed first, the answer is 20. 4. If you add first, the answer is 49. If you square the 3 and then square the 4 (which the problem does not say to do!), then the answer would be 25. (IMPORTANT--SEE NEXT PARAGRAPH
FOR CONCLUSION !
! ) A mathematics system must have unique solutions to basic operations such as these. In order to perform these operations and always arrive at the same conclusion, we must agree to certain traditions which are called the order of operations agreement. This agreement is generally accepted across the world and in most arenas of life. If you obey these traditions, you will obtain consistent results in agreement with everyone else.
Now, we can say of the examples on the previous page, the “correct answers” are:
1.
16;
2. 4;
3. 20;
4. 49 EXERCISES: Simplify each of the following according to
the order of operations agreement. 1.
2 + 8 · 4 2.
20 ÷ 4 ·
5
3. 2 · 5 2
=
__________ = __________
= __________ =
__________ = __________
= __________ 4.
32 ÷ 4 ·
8
5. 100 ÷ 5 2
6. (2 · 5) 2
=
__________ = __________
= __________ =
__________ = __________
= __________ 7.
12 + 8 ÷ 2
8. 12 ÷ (3 + 3)
9. 5 + 9 · 2
= __________ = __________
= __________
= __________ = __________
= __________ 10.
(100 ÷ 5) 2
11. 12 ÷ 3
+ 3 12.
(5 + 9) · 2
= __________ = __________
= __________
=
__________
= __________
= __________ 13.
20 - 5 · 0
14.
12 + 3 ÷ 3
15. 12 ÷ (3 ÷ 3)
= __________ = __________
= __________
= __________ = __________
= __________ 16.
5 + 5 2 17.
(12 + 3) ÷ 3 18.
18 + 2 · 0
= __________ = __________ =
_________ = __________ = __________
= __________ 19.
8 + 3 · 2 2
20. 5 · 3 2 + 11 – 2 2
= _______________ =
_______________
= _______________ =
_______________
= _______________ =
_______________
21.
24 - 3 · 4 + 6 ÷ 2 22.
12 + 18 ÷ 3 2
+
6
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
23.
12 + (18 ÷ 3)2
+ 6
24. 6 + 6 2
÷ 3 · 2
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________ 25.
(16 + 22 ) ÷ 2 ·
2
26. 8 + 2 · 3 2 - 5
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________ 27.
52 + 42 ÷ 22 + 32
28. (16 + 2)2 ÷
(2 + 2)
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________
= _______________ = _______________ 29.
3·42 + 7 - 2·32
÷ 6 + 3·5 2
30. 24 - 12 ÷ 2 · 3 + 6 ·
2 3
= _________________________
= __________________________ =
_________________________
= ________________________
= _________________________
= __________________________ =
_________________________
= __________________________ =
_________________________
= __________________________ 31.
(5 · 2)2 - 20 ÷ 2 ·
(8 - 3) + 10 - 3 + 7
32.
35 - 20 ÷ 5 + 72 · 2 - 6 · 3 + 9 + 10 ÷ 2
33.
35.
36.
SIGNED
NUMBERS Frequently
in algebra you are required to simplify expressions with negative as
well as positive numbers. When
adding numbers it is best to think in terms of MONEY $$$$$!
A positive number is like money coming in to your
possession, or income; a negative number is like money going out,
like expenditures or debts. When
writing a negative number, it is helpful to write the negative number in
parentheses. For
example, 8 + (-5) means you have 8 dollars in your possession and you
spend 5 dollars. The result
is +3 or 3, which means you have 3 dollars. What
if the larger “amount” is negative?
For example, 5+(-12) means you have 5 dollars and spend 12
dollars. The result is (-7)
or a debt of 7 dollars. What
if both “amounts” are negative?
If you have two debts, like (-7) + (-12), the result is (-19) or
a total debt of 19 dollars. In
summary, it is obvious that when you add positive numbers, you get a
positive number. When you
add positive and negative numbers, you subtract the numbers and the sign
of the answer is the same as the sign of the larger “magnitude.”
When you add negative numbers, you always add the numbers and the
sign is always negative.
EXERCISES: 1.
(-12) + 20 = ____
2. (-20) + 12 = ____
3. (-12) + (-8) = ____ 4. 16 + (-4) =
____
5. (-34) + (-16) = ____
6. (-14) + (-28) = ____ 7. (-125) + 28 = ____
8. (-52) + 120 = ____
9. (-12) + (-87) = ____ 10.
(-12) + 34 + (-26) = _______
11. (-28) + (-125) + 95 = _______ 12. 200 + (-120) + (-85) = ______ 13. (-200) + (-135) + 75 = ______ When
subtracting negative numbers, remember that the negative of a negative is
the positive of the number. For examples, -(-8) is a (+8) or just 8; -(-10) is +10 or 10;
-(-x) is x; etc. Of course,
the negative of a positive is a negative.
As examples, -(+8)
is (-8); ‑(+10) is -10; etc.
EXERCISES: 14. -(-12) = _____
15. -(-18) = _____
16. -(-120) = _______ 17.
8 - (-12) = 8 + ___
18. -8 - (-12) = -8 + ____ =
________
= _________ 19.
12 - (-8) = ________
20. -12 - (-8) = _________
= ________
= _________ 21.
(-38) - (-12) = ________
22. 38 - (-12) = _________
= ________
= _________ When
multiplying or dividing positive and negative numbers, remember that
multiplying is actually a short way to add.
For example, 4 x 3 or 4 • 3 means 4 threes.
In this way 4 • (-3) means 4 three dollar debts or (-12).
Likewise, (-3) • 4 is also (-12).
In the same way that a negative
of a negative is a positive, a negative
times a negative is a positive.
The following are rules for multiplication and division of signed
numbers. Remember also, that
the word “of” means “times”.
EXERCISES: 23.
7(-8) = ______
24. (-6)(-9) = _____
25. (-23) 4 = _____ 26.
(-17)(-3) = ______
27. (-16)(9) = _____ 28.
(-5) (-24) = ____ 29.
7(-8)(-2) = _____ 30.
5(-9)(2) = _____ 31.
(-2)(-4)(-25) = _____ 32.
(-7)(-8)(-2) = ____
33. 5(-9)(-2) = _____
34. (-14)(-4)(-25) = ____ When
raising to a power, remember that a positive number raised to any power is
automatically positive. A negative number raised
to an even power is always positive, and a negative number to an odd power
is negative.
EXERCISES: 35.
(-2)2
= _____
36.
(-5)2
= _____
37.
(-4)2
= _____ 38.
(-5)3
= _____
39.
(-3)3
= _____
40.
(-4)3
= _____ 41.
(-1)12
= _____
42.
(-1)9
= _____
43.
(-1)15
= _____ 44.
(-1)24
= _____
45.
(-10)3
= _____
46.
(-3)4
= _____ 47.
(-2)3(-1)
= ____
48.
(-3)4
(-1)8=
____
49.
(-2)4
(-1)3
= ____ 50.
(-2)5(-1)2
= ____
51.
(-5)2
(-2)3
=
____
52.
(-3)2
(-2)3
=
____ 53. However, be
careful when the negative number does not have parentheses.
Is there a difference between
(-2)2 and -2 2 ? The quantity (-2)2 means (-2)
times (-2), which is +4. However,
-22 means the negative of (two to the second power).
By order of operations agreement, this means to raise two to the
second power, and then take the negative.
This result is -4. The important question to ask is this: “What is it that you
are raising to the power?” In
the case of (-2)2 , you are
raising (-2) to the second power. However,
with -2 2 only the 2 is squared, not the “-”.
Therefore, (-2)2 = 4, but -22 = -4 Also
notice that in (-22), only the 2 (not the negative) is squared. Therefore
(-22)= -4. Notice that
(-2)2
(-22 ). EXERCISES. 56.
- 24 = _____
57. (-2)4 = _____
58. (-5)2 = _____ 59.
- 52 = _____
60. (-2)3 = _____
61. -23 = _____
62.
- 34 = _____
63. (-3)4 = _____
64. (-5)3 = _____
65.
- 53 = _____
66. - 110 = _____
67. -113 = _____ 68.
- 22 (-3)3
=
69. - 23 (-3)2 = 70. (-3)3 (-22)= 71.
(-22 )(-33
)=
72.
(-23)(-32
)=
73.
(-33 )(-22 )= 74. (-1)2 (-23 )= 75. (-23 )(-12 )= 76. (-13 )(-22 )= ABSOLUTE VALUE The absolute
value of a number, denoted by vertical bars on each side of a number
or a quantity, represents the size or
magnitude of that number.
Another way to think of absolute value of a number is the distance
on the number line of that number from zero.
As examples, |3| is 3; |7| is 7; |10| is 10; |0| is 0. Notice
that the absolute value definition does not apply to what is outside the
absolute value bars. For
example, -|3| is -3; -|3| is -3; -|7| is -7; -|7| is -7. EXERCISES: 77.
|4| + 3|3|
78. |5| - 3|3|
79.
-|8| + 3|9|
80.
-3| 5| - 5| - 6| 81.
–4 |4 6| - 8| 8 + 3|
82.
9|7+1| - 3|5 9|
83.
|-11|
- | - 5| 2 84.
|-5 - 3| 2
- 4|7 - 2|
85. | - 8 - 5| 2
+ |3 - 12| 2 86.
|-8
- 5| + |3 - 12|
87.
|-8 + 5| - |3 - 12|
88.
- | - 5 - 3| 89.
91.
ANSWERS
1.02 p.15-19:
1. 34;
2. 25; 3. 50; 4. 64; 5.
4; 6. 100; 7. 16; 8.
2; 9. 23; 10. 400;
11. 7;
12. 28; 13. 20; 14. 13; 15.
12; 16. 30; 17. 5; 18.
18; 19. 20; 20. 52; 21.
15; 22. 20; 23. 54;
24. 30; 25. 20; 26. 21; 27.
38; 28. 81; 29. 127; 30.
54; 31. 64; 32. 125; 33.
2; 34. 1; 35. 5; 36.
4. p.21-26:
1. 8;
2. -8; 3. -20; 4. 12; 5.
-50; 6. -42; 7. -97; 8. 68;
9. -99;
10. -4; 11. -58; 12. -5;
13. -260; 14. 12; 15. 18; 16.
120; 17. 20; 18. 4;
19. 20;
20. -4; 21. -26; 22.
50;
23. -56; 24. 54;
25. -92; 26. 51; 27. ‑144;
28. 120;
29. 112; 30. -90;
31. -200;
32. -112; 33. 90; 34.
-1400; 35. 4; 36. 25; 37. 16; 38. -125;
39. -27; 40. -64; 41. 1;
42. -1; 43. -1; 44. 1; 45. ‑1000;
46. 81; 47.
8;
48. 81;
49. -16;
50. -32;
51. ‑200;
52. -72; 53. -108;
54. 216; 55. 36; 56. -16; 57.
16; 58. 25; 59. -25; 60.
-8; 61. -8; 62. -81;
63. 81;
64. -125;
65. -125;
66. -1;
67. -1;
68. 108;
69. -72;
70. 108;
71. 108;
72. 72; 73. 108; 74. -8; 75.
8; 76. 4; 77. 13; 78.
-4; 79. 19; 80. -45 81.
-48: 82. 42; 83. -14;
84. 44; 85. 250; 86. 72; 87.
56; 88. -34: 89. 9; 90.
-2; 91. -2; 92. 2. Return to main page Math in Living C O L O R !! Return to Basic Algebra page Return to Intermediate Algebra page Return to College Algebra page
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