Order of Operations

1.2. ORDER OF OPERATIONS
1.2
7
Order of Operations
1. The Rules Guiding Order of Operations require that multiplication is applied first, then addition.
−12 + 6(−4) = −12 + (−24)
= −36
Multiply first: 6(−4) = −24.
Add: −12 + (−24) = −36.
3. In the expression −(−2)5 , the exponent 5 tells us to write the base −2 five
times as a factor. Thus,
−(−2)5 = −(−2)(−2)(−2)(−2)(−2)
Multiply: (−2)(−2)(−2)(−2)(−2) = −32.
−(−2)5 = −(−32)
Finally, take the opposite.
−(−2)5 = 32
5. First take the absolute value, then negate (take the opposite) of the result.
−| − 40| = −(40)
= −40
Absolute value: | − 40| = 40
Negate.
7. The Rules Guiding Order of Operations require that division and multiplication must be done in the order that they appear, working from left to
right.
−24/(−6)(−1) = 4(−1)
= −4
Divide: −24/(−6) = 4.
Multiply: 4(−1) = −4.
9. The opposite of −50 is 50, or if you wish, the negative of −50 is 50. That
is,
−(−50) = 50.
11. In the expression −35 , the exponent 5 tells us to write the base 3 five times
as a factor. Thus,
−35 = −(3)(3)(3)(3)(3).
Note that order of operations forces us to apply the exponent before applying
the minus sign.
−35 = −243
Second Edition: 2012-2013
8
CHAPTER 1. THE ARITHMETIC OF NUMBERS
13. The Rules Guiding Order of Operations require division and multiplication
must be done in the order that they appear, working from left to right.
48 ÷ 4(6) = 12(6)
= 72
Divide: 48 ÷ 4 = 12.
Multiply: 12(6) = 72.
15.
−52 − 8(−8) = −52 − (−64)
= −52 + 64
= 12
Multiply first: 8(−8) = −64.
Add the opposite.
Add: −52 + 64 = 12.
17. In the expression (−2)4 , the exponent 4 tells us to write the base −2 four
times as a factor. Thus,
(−2)4 = (−2)(−2)(−2)(−2).
Multiply. An even number of negative signs yield a positive result.
(−2)4 = 16
19. The Rules Guiding Order of Operations require that we address exponents
first, then multiplications, then subtractions.
9 − 3(2)2 = 9 − 3(4)
Exponent first: (2)2 = 4.
= 9 − 12
Multiply: 3(4) = 12.
= 9 + (−12)
= −3
Add the opposite.
Add: 9 + (−12) = −3.
21. We must first evaluate the expression inside the absolute value bars. Subtraction means “add the opposite.”
17 − 10|13 − 14| = 17 − 10|13 + (−14)|
= 17 − 10| − 1|
Add the opposite.
Add: 13 + (−14) = −1
= 17 − 10(1)
= 17 − 10
Absolute value: | − 1| = 1
Multiply: 10(1) = 10
= 17 + (−10)
=7
Add the opposite.
Add: 17 + (−10) = 7
Second Edition: 2012-2013
1.2. ORDER OF OPERATIONS
9
23. The Rules Guiding Order of Operations require that we address exponents
first, then multiplications, then subtractions.
−4 + 5(−4)3 = −4 + 5(−64)
= −4 + (−320)
Exponent first: (−4)3 = −64.
Multiply: 5(−64) = −320.
= −324
Add: −4 + (−320) = −324.
25. The Rules Guiding Order of Operations require evaluate the expression
inside the parentheses first, then multiply, then subtract.
8 + 5(−1 − 6) = 8 + 5(−1 + (−6))
= 8 + 5(−7)
Add the opposite.
Add:−1 + (−6) = −7.
Multiply: 5(−7) = −35.
Add: 8 + (−35) = −27.
= 8 + (−35)
= −27
27. Following the Rules Guiding Order of Operations, we must first simplify
the expressions inside the parentheses. Then we can apply the exponents and
after that, subtract.
(10 − 8)2 − (7 − 5)2
= 22 − 23
Subtract: 10 − 8 = 2; 7 − 5 = 2.
=4−8
Square: 22 = 4.
Cube: 23 = 8.
= 4 + (−8)
= −4
Add the opposite.
Add: 4 + (−8) = −4.
29. The Rules Guiding Order of Operations require evaluate the expression
inside the innermost parentheses first.
6 − 9(6 − 4(9 − 7)) = 6 − 9(6 − 4(9 + (−7)));
Add the opposite.
= 6 − 9(6 − 4(2));
= 6 − 9(6 − 8);
Add: 9 + (−7) = 2.
Multiply: 4(2) = 8.
= 6 − 9(6 + (−8));
= 6 − 9(−2);
Add the opposite.
Add: 6 + (−8) = −2.
= 6 − (−18);
= 6 + 18;
Multiply: 9(−2) = −18.
Add the opposite.
= 24
Add: 6 + 18 = 24.
Second Edition: 2012-2013
10
CHAPTER 1. THE ARITHMETIC OF NUMBERS
31. The Rules Guiding Order of Operations require evaluate the expression
inside the parentheses first, then multiply, then subtract.
−6 − 5(4 − 6) = −6 − 5(4 + (−6))
Add the opposite.
= −6 − 5(−2)
= −6 − (−10)
Add:4 + (−6) = −2.
Multiply: 5(−2) = −10.
= −6 + 10
=4
Add the opposite.
Add: −6 + 10 = 4.
33. Following the Rules Guiding Order of Operations, simplify the expression
inside the parentheses first, then apply the exponent, then add and subtract,
moving left to right.
9 + (9 − 6)3 − 5 = 9 + 33 − 5
Subtract: 9 − 6 = 3.
= 9 + 27 − 5
= 9 + 27 + (−5)
Exponent: 33 = 27.
Add the opposite.
= 36 + (−5)
= 31
Add: 9 + 27 = 36.
Add: 36 + (−5) = 31.
35. The Rules Guiding Order of Operations require that we address exponents
first, then multiplications, then subtractions.
−5 + 3(4)2 = −5 + 3(16)
Exponent first: (4)2 = 16.
= −5 + 48
Multiply: 3(16) = 48.
= 43
Add: −5 + 48 = 43.
37. Following the Rules Guiding Order of Operations, simplify the expression
inside the parentheses first, then apply the exponent, then add and subtract,
moving left to right.
8 − (5 − 2)3 + 6 = 8 − 33 + 6
Subtract: 5 − 2 = 3.
= 8 − 27 + 6
Exponent: 33 = 27.
= 8 + (−27) + 6
= −19 + 6
Add the opposite.
Add: 8 + (−27) = −19.
= −13
Add: −19 + 6 = −13.
Second Edition: 2012-2013
1.2. ORDER OF OPERATIONS
11
39. We must first evaluate the expression inside the absolute value bars. Start
with “subtraction means add the opposite.”
|6 − 15| − | − 17 − 11|
= |6 + (−15)| − | − 17 + (−11)|
= | − 9| − | − 28|
= 9 − 28
= 9 + (−28)
= −19
Add the opposites.
Add: 6 + (−15) = −9
and −17 + (−11) = −28
Absolute value: | − 9| = 9
and | − 28| = 28
Add the opposite.
Add: 9 + (−28) = −19
41. The Rules Guiding Order of Operations require evaluate the expression
inside the innermost parentheses first.
5 − 5(5 − 6(6 − 4)) = 5 − 5(5 − 6(6 + (−4)));
Add the opposite.
= 5 − 5(5 − 6(2));
= 5 − 5(5 − 12);
Add: 6 + (−4) = 2.
Multiply: 6(2) = 12.
= 5 − 5(5 + (−12));
= 5 − 5(−7);
Add the opposite.
Add: 5 + (−12) = −7.
= 5 − (−35);
= 5 + 35;
Multiply: 5(−7) = −35.
Add the opposite.
= 40
Add: 5 + 35 = 40.
43. First replace all occurrences of the variables in the expression with open
parentheses:
4x2 + 3xy + 4y 2 = 4( )2 + 3( )( ) + 4( )2
Then replace each variable with its given value, and evaluate the expression:
4x2 + 3xy + 4y 2 = 4(−3)2 + 3(−3)(0) + 4(0)2
= 4(9) + 3(−3)(0) + 4(0)
= 36
= 36
Substitute −3 for x and 0 for y.
Evaluate exponents first.
Perform multiplications, left to right.
Perform additions and subtractions, left to right.
45. First replace all occurrences of the variables in the expression with open
parentheses:
−8x + 9 = −8( ) + 9
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
12
Then replace each variable with its given value, and evaluate the expression:
−8x + 9 = −8(−9) + 9
= 72 + 9
= 81
Substitute −9 for x.
Multiply first: −8(−9) = 72
Add.
47. First replace all occurrences of the variables in the expression with open
parentheses:
−5x2 + 2xy − 4y 2 = −5( )2 + 2( )( ) − 4( )2
Then replace each variable with its given value, and evaluate the expression:
−5x2 + 2xy − 4y 2 = −5(5)2 + 2(5)(0) − 4(0)2
= −5(25) + 2(5)(0) − 4(0)
= −125
= −125
Substitute 5 for x and 0 for y.
Evaluate exponents first.
Perform multiplications, left to right.
Perform additions and subtractions, left to right.
49. First replace all occurrences of the variables in the expression with open
parentheses:
3x2 + 3x − 4 = 3( )2 + 3( ) − 4
Then replace each variable with its given value, and evaluate the expression:
3x2 + 3x − 4 = 3(5)2 + 3(5) − 4
Substitute 5 for x.
= 3(25) + 3(5) − 4
= 75 + 15 − 4
Evaluate exponents first.
Perform multiplications, left to right.
= 86
Perform additions and subtractions, left to right.
51. First replace all occurrences of the variables in the expression with open
parentheses:
−2x2 + 2y 2 = −2( )2 + 2( )2
Then replace each variable with its given value, and evaluate the expression:
−2x2 + 2y 2 = −2(1)2 + 2(−2)2
Substitute 1 for x and −2 for y.
= −2(1) + 2(4)
= −2 + 8
Evaluate exponents first.
Perform multiplications, left to right.
=6
Perform additions and subtractions, left to right.
Second Edition: 2012-2013
1.2. ORDER OF OPERATIONS
13
53. First replace all occurrences of the variables in the expression with open
parentheses:
−3x2 − 6x + 3 = −3( )2 − 6( ) + 3
Then replace each variable with its given value, and evaluate the expression:
−3x2 − 6x + 3 = −3(2)2 − 6(2) + 3
= −3(4) − 6(2) + 3
= −12 − 12 + 3
= −21
Substitute 2 for x.
Evaluate exponents first.
Perform multiplications, left to right.
Perform additions and subtractions, left to right.
55. First replace all occurrences of the variables in the expression with open
parentheses:
−6x − 1 = −6( ) − 1
Then replace each variable with its given value, and evaluate the expression:
−6x − 1 = −6(1) − 1
= −6 − 1
= −7
Substitute 1 for x.
Multiply first: −6(1) = −6
Subtract.
57. First replace all occurrences of the variables in the expression with open
parentheses:
3x2 − 2y 2 = 3( )2 − 2( )2
Then replace each variable with its given value, and evaluate the expression:
3x2 − 2y 2 = 3(−3)2 − 2(−2)2
Substitute −3 for x and −2 for y.
= 3(9) − 2(4)
= 27 − 8
Evaluate exponents first.
Perform multiplications, left to right.
= 19
Perform additions and subtractions, left to right.
59. Replace each occurrence of the variables a and b with open parentheses,
then substitute 27 for a and −30 for b.
2
2
a2 + b 2
( ) +( )
=
a+b
( )+( )
2
=
Replace variables with open parentheses.
2
(27) + (−30)
(27) + (−30)
Substitute: 27 for a and −30 for b.
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
14
In the numerator, exponents first, then add. In the denominator, add. Finally,
divide numerator by denominator.
=
729 + 900
−3
Numerator, exponents first.
Denominator, add.
=
1629
−3
Numerator: 729 + 900 = 1629
= −543
Divide:
1629
= −543
−3
61. Replace each occurrence of the variables a, b, c, and d with open parentheses, then substitute −42 for a, 25 for b, 26 for c, and 43 for d.
( )+( )
a+b
=
c−d
( )−( )
(−42) + (25)
=
(26) − (43)
Replace variables with open parentheses.
Substitute: −42 for a, 25 for b,
26 for c, and 43 for d.
In the denominator, change the subtraction to adding the opposite. Next,
simplify numerator and denominator, then divide.
−42 + 25
26 + (−43)
−17
=
−17
=
=1
In the denominator, add the opposite.
Numerator: −42 + 25 = −17
Denominator: 26 + (−43) = −17
−17
=1
Divide:
−17
63. Replace each occurrence of the variables a, b, c, and d with open parentheses, then substitute −7 for a, 48 for b, 5 for c, and 11 for d.
a−b
( )−( )
=
cd
( )( )
(−7) − (48)
=
(5) (11)
Replace variables with open parentheses.
Substitute: −7 for a, 48 for b,
5 for c, and 11 for d.
In the numerator, change the subtraction to adding the opposite. Next, simplify
numerator and denominator, then divide.
−7 + (−48)
(5)(11)
−55
=
55
=
= −1
Second Edition: 2012-2013
In the numerator, add the opposite.
Numerator: −7 + (−48) = −55
Denominator: (5)(11) = 55
−55
Divide:
= −1
55
1.2. ORDER OF OPERATIONS
15
65. Following Tips for Evaluating Algebraic Expressions, first replace all occurrences of variables in the expression a2 + b2 with open parentheses, then
substitute 3 for a and 4 for b and simplify.
a2 + b2 = ( )2 + ( )2
2
Replace variables with open parentheses.
2
= (3) + (4)
Substitute: 3 for a, 4 for b.
= 9 + 16
Square: (3)2 = 9; (4)2 = 16.
= 25
Add: 9 + 16 = 25.
Now we deal with the second expression.
(a + b)2 = (( ) + ( ))2
Replace variables with open parentheses.
2
Substitute: 3 for a, 4 for b.
= ((3) + (4))
2
= (7)
Add: 3 + 4 = 7.
= 49
Square: (7)2 = 49.
Thus, if a = 3 and b = 4, we found that a2 + b2 = 25, but (a + b)2 = 49. Hence
the expressions a2 + b2 and (a + b)2 did not produce the same results.
67. Following Tips for Evaluating Algebraic Expressions, first replace all occurrences of variables in the expression |a||b| with open parentheses, then substitute −3 for a and 5 for b and simplify.
|a||b| = |( )||( )|
Replace variables with open parentheses.
= |(−3)||(5)|
= (3)(5)
Substitute: −3 for a, 5 for b.
Simplify: | − 3| = 3; |5| = 5.
= 15
Multiply: (3)(5) = 15.
Now we deal with the second expression.
|ab| = |( )( )|
Replace variables with open parentheses.
= |(−3)(5)|
= | − 15|
Substitute: −3 for a, 5 for b.
Multiply: (−3)(5) = −15.
= 15
Simplify: | − 15| = 15.
Thus, if a = −3 and b = 5, we found that |a||b| = 15 and |ab| = 15. Hence the
expressions |a||b| and |ab| did produce the same results.
69. To evaluate the expression −236 − 324(−576 + 57), enter the expression
-236-324*(-576+57) using the following keystrokes.
(-)
2
3
6
−
3
2
4
×
(
(-)
5
7
6
Second Edition: 2012-2013
CHAPTER 1. THE ARITHMETIC OF NUMBERS
16
+
5
)
7
ENTER
The result is shown in the following figure.
Hence, −236 − 324(−576 + 57) = 167920.
71. To evaluate the expression
using the following keystrokes.
(
2
7
−
0
−
270 − 900
, enter the expression (270-900)/(300-174)
300 − 174
9
1
0
7
)
4
÷
)
0
(
3
0
0
ENTER
The result is shown in the following figure.
Hence,
270 − 900
= −5.
300 − 174
73. First, store −93 in A with the following keystrokes. The variable A is
located above the MATH button. Press the ALPHA key to access A.
(-)
9
3
STO
ALPHA
A
Second, store 84 in B with the following keystrokes. The variable B is located
above the APPS button. Press the ALPHA key to access B.
8
Second Edition: 2012-2013
4
STO
ALPHA
B
1.2. ORDER OF OPERATIONS
17
Finally, enter the expression (A^2+B^2)/(A+B) with the following keystrokes.
(
ALPHA
(
A
∧
ALPHA
+
2
A
+
ALPHA
ALPHA
∧
B
B
)
2
)
÷
ENTER
The results are shown in the following figure
Hence,
a2 + b 2
= −1745.
a+b
75. Start with the formula F = (9/5)C + 32, replace the variables C with open
parentheses, then substitute 60 for C and simplify.
9
Fahrenheit formula.
F = C + 32
5
9
Replace C with open parentheses.
= ( ) + 32
5
9
= (60) + 32
Substitute: 60 for C.
5
= 108 + 32
Multiply: (9/5)60 = 108.
= 140
Add: 108 + 32 = 140.
Hence, the Fahrenheit temperature is 140◦ F.
77. Start with the kinetic energy formula K = (1/2)mv 2 , replace the variables
m and v with open parentheses, then substitute 7 for m and 50 for v and
simplify.
1
Kinetic energy formula.
K = mv 2
2
1
Replace m and v with open parentheses.
= ( )( )2
2
1
= (7)(50)2
Substiute: 7 for m, 50 for v.
2
1
Square: (50)2 = 2500.
= (7)(2500)
2
= 8750
Multiply: (1/2)(7)(2500) = 8750.
Hence, the kinetic energy of the object is 8, 750 joules.
Second Edition: 2012-2013