16
CHAPTER 1. THE ARITHMETIC OF NUMBERS
1.2
Order of Operations
The order in which we evaluate expressions can be ambiguous. Take, for example, the expression −4 + 2 · 8. If we perform the addition first, then we get
−16 as a result (the question mark over the equal sign indicates that the result
is questionable).
?
−4 + 2 · 8 = −2 · 8
?
= −16.
On the other hand, if we perform the multiplication first, then we get 12 as a
result.
?
−4 + 2 · 8 = −4 + 16
?
= 12.
So, what are we to do?
Of course, grouping symbols would remove the ambiguity.
Grouping Symbols. Parentheses, brackets, and absolute value bars can be
used to group parts of an expression. For example:
3 + 5(9 − 11)
or
− 2 − [−2 − 5(1 − 3)]
or
6 − 3| − 3 − 4|
In each case, the rule is “evaluate the expression inside the grouping symbols first.” If the grouping symbols are nested, evaluate the expression in the
innermost pair of grouping symbols first.
Thus, if the example above is grouped as follows, we are forced to evaluate the
expression inside the parentheses first.
(−4 + 2) · 8 = −2 · 8
= −16
Parentheses first: −4 + 2 = −2
Multiply: −2 · 8 = −16
Another way to avoid ambiguities in evaluating expressions is to establish
an order in which operations should be performed. The following guidelines
should always be strictly enforced when evaluating expressions.
Rules Guiding Order of Operations. When evaluating expressions, proceed in the following order.
1. Evaluate expressions contained in grouping symbols first. If grouping
symbols are nested, evaluate the expression in the innermost pair of
grouping symbols first.
17
1.2. ORDER OF OPERATIONS
2. Evaluate all exponents that appear in the expression.
3. Perform all multiplications and divisions in the order that they appear
in the expression, moving left to right.
4. Perform all additions and subtractions in the order that they appear in
the expression, moving left to right.
You Try It!
EXAMPLE 1. Simplify: −3 − 4 · 8
Simplify:
−4 + 2 · 8
Solution: Because of the established Rules Guiding Order of Operations, this
expression is no longer ambiguous. There are no grouping symbols or exponents
present, so we immediately go to rule three, evaluate all multiplications and
divisions in the order that they appear, moving left to right. After that we
invoke rule four, performing all additions and subtractions in the order that
they appear, moving left to right.
−3 − 4 · 8 = −3 − 32
Multiply first: 4 · 8 = 32
= −3 + (−32)
Add the opposite.
= −35
Add: −3 + (−32) = −35
Thus, −3 − 4 · 8 = −35.
Answer: 12
Writing Mathematics. When simplifying expressions, observe the following
rule to neatly arrange your work:
One equal sign per line. This means that you should not arrange your work
horizontally.
−2 − 4 · (−8) = −2 − (−32) = −2 + 32 = 30
That’s three equal signs on a single line. Rather, arrange your work vertically,
keeping equal signs aligned in a column.
−2 − 4 · (−8) = −2 − (−32)
= −2 + 32
= 30
18
CHAPTER 1. THE ARITHMETIC OF NUMBERS
You Try It!
Simplify:
−24/(−3)(2)
EXAMPLE 2. Simplify: 54/(−9)(2)
Solution: There are no grouping symbols or exponents present, so we immediately go to rule three, evaluate all multiplications and divisions in the order
that they appear, moving left to right.
54/(−9)(2) = −6(2)
Divide first: 54/(−9) = −6
= −12
Answer: 16
Multiply: −6(2) = −12
Thus, 54/(−9)(2) = −12.
Example 2 can be a source of confusion for many readers. Note that multiplication takes no preference over division, nor does division take preference over
multiplication. Multiplications and divisions have the same level of preference
and must be performed in the order that they occur, moving from left to right.
If not, the wrong answer will be obtained.
Warning! Here is what happens if you perform the multiplication in Example 2 before the division.
54/(−9)(2) = 54/(−18)
= −3
Multiply: (−9)(2) = −18
Divide: 54/(−18) = −3
This is incorrect! Multiplications and divisions must be performed in the
order that they occur, moving from left to right.
You Try It!
Simplify:
−152
EXAMPLE 3. Simplify:
(a) (−7)2 and (b) −72
Solution. Recall that for any integer a, we have (−1)a = −a. Because negating is equivalent to multiplying by −1, the Rules Guiding Order of Operations
require that we address grouping symbols and exponents before negation.
a) Because of the grouping symbols, we negate first, then square. That is,
(−7)2 = (−7)(−7)
= 49.
b) There are no grouping symbols in this example. Thus, we must square first,
then negate. That is,
−72 = −(7 · 7)
= −49.
Answer: −225
Thus, (−7)2 = 49, but −72 = −49. Note: This example demonstrates that
(−7)2 is different from −72 .
19
1.2. ORDER OF OPERATIONS
Let’s try an example that has a mixture of exponents, multiplication, and
subtraction.
You Try It!
EXAMPLE 4. Simplify:
−3 − 2(−4)2
Simplify:
−5 − 4(−2)3
Solution. The Rules Guiding Order of Operations require that we address
exponents first, then multiplications, then subtractions.
−3 − 2(−4)2 = −3 − 2(16)
Exponent first: (−4)2 = 16
= −3 − 32
Multiply: 2(16) = 32
= −3 + (−32)
= −35
Add the opposite.
Add: −3 + (−32) = −35
Thus, −3 − 2(−4)2 = −35.
Answer: 27
Grouping Symbols
The Rules Guiding Order of Operations require that expressions inside grouping symbols (parentheses, brackets, or curly braces) be evaluated first.
You Try It!
EXAMPLE 5. Simplify:
−2(3 − 4)2 + 5(1 − 2)3
Simplify:
−2 − 3(−2 − 3)3
Solution. The Rules Guiding Order of Operations require that we first evaluate the expressions contained inside the grouping symbols.
−2(3 − 4)2 + 5(1 − 2)3
= −2(3 + (−4))2 + 5(1 + (−2))3
2
3
= −2(−1) + 5(−1)
Add the opposites.
Parentheses first: 3 + (−4) = −1
and 1 + (−2) = −1.
Evaluate the exponents next, perform the multiplications, then add.
= −2(1) + 5(−1)
= −2 + (−5)
= −7
Thus, −2(3 − 4)2 + 5(1 − 2)3 = −7.
Exponents: (−1)2 = 1
and (−1)3 = −1.
Multiply: −2(1) = −2
and 5(−1) = −5.
Add: −2 + (−5) = −7
Answer: 373
20
CHAPTER 1. THE ARITHMETIC OF NUMBERS
Absolute Value Bars as Grouping Symbols
Like parentheses and brackets, you must evaluate what is inside them first,
then take the absolute value of the result.
You Try It!
Simplify: −| − 4 − 6|
EXAMPLE 6. Simplify:
−8 − |5 − 11|
Solution. We must first evaluate what is inside the absolute value bars.
−8 − |5 − 11| = −8 − |5 + (−11)|
= −8 − | − 6|
Add the opposite.
Add: 5 + (−11) = −6.
The number −6 is 6 units from zero on the number line. Hence, | − 6| = 6.
Answer: −10
= −8 − 6
= −8 + (−6)
Add: | − 6| = 6.
Add the opposite.
= −14
Add.
Thus, −8 − |5 − 11| = −14.
Nested Grouping Symbols
When grouping symbols are nested, the Rules Guiding Order of Operations tell
us to evaluate the innermost expressions first.
You Try It!
Simplify:
−2 − 2[−2 − 2(−2 − 2)]
EXAMPLE 7. Simplify:
−3 − 4[−3 − 4(−3 − 4)]
Solution. The Rules Guiding Order of Operations require that we first address the expression contained in the innermost grouping symbols. That is, we
evaluate the expression contained inside the brackets first.
−3 − 4[−3 − 4(−3 − 4)] = −3 − 4[−3 − 4(−3 + (−4))] Add the opposite.
= −3 − 4[−3 − 4(−7)]
Add: −3 + (−4) = −7
Next, we evaluate the expression contained inside the brackets.
= −3 − 4[−3 − (−28)]
= −3 − 4[−3 + 28]
Multiply: 4(−7) = −28
Add the opposite.
= −3 − 4[25]
Add: −3 + 28 = 25
Now we multiply, then subtract.
21
1.2. ORDER OF OPERATIONS
= −3 − 100
Multiply: 4{25} = 100
= −3 + (−100)
= −103
Add the opposite.
Add: −3 + (−100) = −103
Thus, −3 − 4[−3 − 4(−3 − 4)] = −103.
Answer: −14
Evaluating Algebraic Expressions
Variable. A variable is a symbol (usually a letter) that stands for an unknown
value that may vary.
Let’s add the definition of an algebraic expression.
Algebraic Expression. When we combine numbers and variables in a valid
way, using operations such as addition, subtraction, multiplication, division,
exponentiation, the resulting combination of mathematical symbols is called
an algebraic expression.
Thus,
2a,
x + 5,
and
y2,
being formed by a combination of numbers, variables, and mathematical operators, are valid algebraic expressions.
An algebraic expression must be well-formed. For example,
2 + −5x
is not a valid expression because there is no term following the plus sign (it is
not valid to write +− with nothing between these operators). Similarly,
2 + 3(2
is not well-formed because parentheses are not balanced.
In this section we will evaluate algebraic expressions for given values of the
variables contained in the expressions. Here are some simple tips to help you
be successful.
Tips for Evaluating Algebraic Expressions.
22
CHAPTER 1. THE ARITHMETIC OF NUMBERS
1. Replace all occurrences of variables in the expression with open parentheses. Leave room between the parentheses to substitute the given value
of the variable.
2. Substitute the given values of variables in the open parentheses prepared
in the first step.
3. Evaluate the resulting expression according to the Rules Guiding Order
of Operations.
You Try It!
If x = −2 and y = −1,
evaluate x3 − y 3 .
EXAMPLE 8. Evaluate the expression x2 − 2xy + y 2 at x = −3 and y = 2.
Solution. Following Tips for Evaluating Algebraic Expressions, first replace all
occurrences of variables in the expression x2 − 2xy + y 2 with open parentheses.
Next, substitute the given values of variables (−3 for x and 2 for y) in the open
parentheses.
x2 − 2xy + y 2 = (
)2 − 2(
)( ) + ( )2
= (−3)2 − 2(−3)(2) + (2)2
Finally, follow the Rules Guiding Order of Operations to evaluate the resulting
expression.
x2 − 2xy + y 2
Original expression.
2
Replace variables with parentheses.
2
= (−3) − 2(−3)(2) + (2)
Substitute −3 for x and 2 for y.
= 9 − 2(−3)(2) + 4
= 9 − (−6)(2) + 4
Evaluate exponents first.
Left to right, multiply: 2(−3) = −6
= 9 − (−12) + 4
Left to right, multiply: (−6)(2) = −12
= 9 + 12 + 4
= 25
Add the opposite.
Add.
=(
2
) − 2(
)( ) + ( )
2
Answer: −7
Thus, if x = −3 and y = 2, then x2 − 2xy + y 2 = 25.
Evaluating Fractions
If a fraction bar is present, evaluate the numerator and denominator separately
according to the Rules Guiding Order of Operations, then perform the division
in the final step.
23
1.2. ORDER OF OPERATIONS
You Try It!
EXAMPLE 9. Evaluate the expression
ad − bc
a+b
at a = 5, b = −3, c = 2, and d = −4.
If a = −7, b = −3, c = −15,
and d = −14, evaluate:
a2 + b 2
c+d
Solution. Following Tips for Evaluating Algebraic Expressions, first replace
all occurrences of variables in the expression (ad − bc)/(a + b) with open parentheses. Next, substitute the given values of variables (5 for a, −3 for b, 2 for c,
and −4 for d) in the open parentheses.
( )( ) − ( )( )
ad − bc
=
a+b
( )+( )
(5)(−4) − (−3)(2)
=
(5) + (−3)
Finally, follow the Rules Guiding Order of Operations to evaluate the resulting expression. Note that we evaluate the expressions in the numerator and
denominator separately, then divide.
ad − bc
( )( ) − ( )( )
=
a+b
( )+( )
(5)(−4) − (−3)(2)
=
(5) + (−3)
−20 − (−6)
=
2
−20 + 6
2
−14
=
2
= −7
=
Replace variables with parentheses.
Substitute: 5 for a, −3 for b, 2 for c, −4 for d
Numerator: (5)(−4) = −20, (−3)(2) = −6
Denominator: 5 + (−3) = 2
Numerator: Add the opposite
Numerator: −20 + 6 = −14
Divide.
Thus, if a = 5, b = −3, c = 2, and d = −4, then (ad − bc)/(a + b) = −7.
Answer: −2
Using the Graphing Calculator
The graphing calculator is a splendid tool for evaluating algebraic expressions,
particularly when the numbers involved are large.
24
CHAPTER 1. THE ARITHMETIC OF NUMBERS
You Try It!
Use the graphing calculator
to evaluate
−2 − 2[−2 − 2(−2 − 2)].
EXAMPLE 10. Use the graphing calculator to simplify the following expression.
−213 − 35[−18 − 211(15 − 223)]
Solution. The first difficulty with this expression is the fact that the graphing
calculator does not have a bracket symbol for the purposes of grouping. The
calculator has only parentheses for grouping. So we first convert our expression
to the following:
−213 − 35(−18 − 211(15 − 223))
Note that brackets and parentheses are completely interchangeable.
The next difficulty is determining which of the minus signs are negation
symbols and which are subtraction symbols. If the minus sign does not appear
between two numbers, it is a negation symbol. If the minus sign does appear
between two numbers, it is a subtraction symbol. Hence, we enter the following
keystrokes on our calculator. The result is shown in Figure 1.8.
(-)
2
1
3
−
3
5
×
(
(-)
1
8
−
1
1
×
(
1
5
−
2
2
3
)
)
ENTER
2
Figure 1.8: Calculating −213 − 35[−18 − 211(15 − 223)].
Answer: −14
Thus, −213 − 35[−18 − 211(15 − 223)] = −1, 535, 663.
You Try It!
Use the graphing calculator
to evaluate
10 + 10
10 + 10
EXAMPLE 11. Use the graphing calculator to evaluate
5+5
5+5
.
25
1.2. ORDER OF OPERATIONS
Solution. You might ask “Why do we need a calculator to evaluate this
exceedingly simple expression?” After all, it’s very easy to compute.
5+5
10
=
5+5
10
=1
Simplify numerator and denominator.
Divide: 10/10 = 1.
Well, let’s enter the expression 5+5/5+5 in the calculator and see how well
we understand the Rules Guiding Order of Operations (see first image in
Figure 1.9). Whoa! How did the calculator get 11? The answer is supposed to
be 1!
Let’s slow down and apply the Rules Guiding Order of Operations to the
expression 5+5/5+5.
5 + 5/5 + 5 = 5 +
5
+5
5
Divide first.
5
= 1.
5
Add: 5 + 1 + 5 = 11.
=5+1+5
Divide:
= 11
Aha! That’s how the calculator got 11.
5 + 5/5 + 5
is equivalent to
5+
5
+5
5
Let’s change the order of evaluation by using grouping symbols. Note that:
Parentheses first.
Divide: 10/10 = 1.
(5 + 5)/(5 + 5) = 10/10
=1
That is:
5+5
5+5
Enter (5+5)(5+5) and press the ENTER key to produce the output shown in
the second image in Figure 1.9.
(5 + 5)/(5 + 5)
is equivalent to
Figure 1.9: Calculating
5+5
.
5+5
Answer: 1
26
CHAPTER 1. THE ARITHMETIC OF NUMBERS
The graphing calculator has memory locations available for “storing” values.
They are lettered A-Z and appear on the calculator case, in alphabetic order
as you move from left to right and down the keyboard. Storing values in
these memory locations is an efficient way to evaluate algebraic expressions
containing variables. Use the ALPHA key to access these memory locations.
You Try It!
Use the graphing calculator
to evaluate |a − b| at
a = −312 and b = −875.
EXAMPLE 12. Use the graphing calculator to evaluate |a| − |b| at a = −312
and b = −875.
Solution. First store −312 in the variable A with the following keystrokes.
To select the letter A, press the ALPHA key, then the MATH key, located in
the upper left-hand corner of the calculator (see Figure 1.10).
(-)
3
1
2
STO.
ALPHA
A
ENTER
Next, store −875 in the variable B with the following keystrokes. To select the
letter B, press the ALPHA key, then the APPS key.
(-)
Figure 1.10: Upper half of the
TI84.
8
7
5
STO.
ALPHA
B
ENTER
The results of these keystrokes are shown in the first image in Figure 1.11.
Now we need to enter the expression |a| − |b|. The absolute value function
is located in the MATH menu. When you press the MATH key, you’ll notice
submenus MATH, NUM, CPX, and PRB across the top row of the MATH
menu. Use the right-arrow key to select the NUM submenu (see the second
image in Figure 1.11). Note that abs( is the first entry on this menu. This
is the absolute value function needed for this example. Enter the expression
abs(A)-abs(B) as shown in the third image in Figure 1.11. Use the ALPHA
key as described above to enter the variables A and B and close the parentheses
using the right parentheses key from the keyboard. Press the ENTER key to
evaluate your expression.
Figure 1.11: Evaluate |a| − |b| at a = −312 and b = −875.
Answer: 563
Thus, |a| − |b| = −563.
27
1.2. ORDER OF OPERATIONS
❧ ❧ ❧
Exercises
❧ ❧ ❧
In Exercises 1-18, simplify the given expression.
1. −12 + 6(−4)
10. −(−30)
2. 11 + 11(7)
11. −35
3. −(−2)5
12. −32
4. −(−5)3
13. 48 ÷ 4(6)
5. −| − 40|
14. 96 ÷ 6(4)
6. −| − 42|
15. −52 − 8(−8)
7. −24/(−6)(−1)
16. −8 − 7(−3)
8. 45/(−3)(3)
17. (−2)4
9. −(−50)
18. (−4)4
In Exercises 19-42, simplify the given expression.
19. 9 − 3(2)2
31. −6 − 5(4 − 6)
20. −4 − 4(2)2
32. −5 − 5(−7 − 7)
21. 17 − 10|13 − 14|
33. 9 + (9 − 6)3 − 5
22. 18 − 3| − 20 − 5|
34. 12 + (8 − 3)3 − 6
23. −4 + 5(−4)3
35. −5 + 3(4)2
24. 3 + 3(−4)3
36. 2 + 3(2)2
25. 8 + 5(−1 − 6)
37. 8 − (5 − 2)3 + 6
26. 8 + 4(−5 − 5)
38. 9 − (12 − 11)2 + 4
27. (10 − 8)2 − (7 − 5)3
39. |6 − 15| − | − 17 − 11|
28. (8 − 10)2 − (4 − 5)3
40. | − 18 − 19| − | − 3 − 12|
29. 6 − 9(6 − 4(9 − 7))
41. 5 − 5(5 − 6(6 − 4))
30. 4 − 3(3 − 5(7 − 2))
42. 4 − 6(4 − 7(8 − 5))
28
CHAPTER 1. THE ARITHMETIC OF NUMBERS
In Exercises 43-58, evaluate the expression at the given values of x and y.
43. 4x2 + 3xy + 4y 2 at x = −3 and y = 0
51. −2x2 + 2y 2 at x = 1 and y = −2
44. 3x2 − 3xy + 2y 2 at x = 4 and y = −3
52. −5x2 + 5y 2 at x = −4 and y = 0
45. −8x + 9 at x = −9
53. −3x2 − 6x + 3 at x = 2
46. −12x + 10 at x = 2
54. −7x2 + 9x + 5 at x = −7
47. −5x2 + 2xy − 4y 2 at x = 5 and y = 0
55. −6x − 1 at x = 1
2
2
48. 3x + 3xy − 5y at x = 0 and y = 3
56. 10x + 7 at x = 9
49. 3x2 + 3x − 4 at x = 5
57. 3x2 − 2y 2 at x = −3 and y = −2
50. 2x2 + 6x − 5 at x = 6
58. −3x2 + 2y 2 at x = 2 and y = 2
59. Evaluate
62. Evaluate
60. Evaluate
63. Evaluate
61. Evaluate
64. Evaluate
a2 + b 2
a+b
at a = 27 and b = −30.
a2 + b 2
a+b
at a = −63 and b = 77.
a+b
c−d
at a = −42, b = 25, c = 26, and d = 43.
a+b
c−d
at a = 38, b = 42, c = 10, and d = 50.
a−b
cd
at a = −7, b = 48, c = 5, and d = 11.
a−b
cd
at a = −46, b = 46, c = 23, and d = 2.
65. Evaluate the expressions a2 + b2 and (a + b)2 at a = 3 and b = 4. Do the expressions produce the
same results?
66. Evaluate the expressions a2 b2 and (ab)2 at a = 3 and b = 4. Do the expressions produce the same
results?
67. Evaluate the expressions |a||b| and |ab| at a = −3 and b = 5. Do the expressions produce the
same results?
68. Evaluate the expressions |a| + |b| and |a + b| at a = −3 and b = 5. Do the expressions produce
the same results?
29
1.2. ORDER OF OPERATIONS
In Exercises 69-72, use a graphing calculator to evaluate the given expression.
69. −236 − 324(−576 + 57)
70. −443 + 27(−414 − 22)
270 − 900
300 − 174
3000 − 952
72.
144 − 400
71.
a2 + b 2
at a = −93 and b = 84 by first storing
a+b
−93 in the variable A and 84 in the variable B, then entering the expression (A^2+B^2)/(A+B).
73. Use a graphing calculator to evaluate the expression
a2 + b 2
at a = −76 and b = 77 by first storing
a+b
−76 in the variable A and 77 in the variable B, then entering the expression (A^2+B^2)/(A+B).
74. Use a graphing calculator to evaluate the expression
75. The formula
the formula
9
C + 32
5
will change a Celsius temperature to a
Fahrenheit temperature. Given that the
Celsius temperature is C = 60◦ C, find the
equivalent Fahrenheit temperature.
F =
76. The surface area of a cardboard box is
given by the formula
S = 2W H + 2LH + 2LW,
where W and L are the width and length
of the base of the box and H is its height.
If W = 2 centimeters, L = 8 centimeters,
and H = 2 centimeters, find the surface
area of the box.
77. The kinetic energy (in joules) of an object
having mass m (in kilograms) and velocity v (in meters per second) is given by
K=
1
mv 2 .
2
Given that the mass of the object is m = 7
kilograms and its velocity is v = 50 meters
per second, calculate the kinetic energy of
the object.
78. The area of a trapezoid is given by the
formula
A=
1
(b1 + b2 ) h,
2
where b1 and b2 are the lengths of the
parallel bases and h is the height of the
trapezoid. If the lengths of the bases are
21 yards and 11 yards, respectively, and if
the height is 22 yards, find the area of the
trapezoid.
30
CHAPTER 1. THE ARITHMETIC OF NUMBERS
❧ ❧ ❧
Answers
❧ ❧ ❧
1. −36
41. 40
3. 32
43. 36
5. −40
45. 81
7. −4
47. −125
9. 50
49. 86
11. −243
13. 72
15. 12
17. 16
19. −3
21. 7
23. −324
51. 6
53. −21
55. −7
57. 19
59. −543
61. 1
63. −1
25. −27
65. No.
27. −4
67. Yes.
29. 24
69. 167920
31. 4
71. −5
33. 31
35. 43
37. −13
39. −19
73. −1745
75. 140◦ F
77. 8750 joules