Download 1.2Evaluate and Simplify Algebraic Expressions

1.2
Before
Now
Why?
Key Vocabulary
• power
• variable
• term
• coefficient
• identity
Evaluate and Simplify
Algebraic Expressions
You studied properties of real numbers.
You will evaluate and simplify expressions involving real numbers.
So you can estimate calorie use, as in Ex. 60.
A numerical expression consists of numbers, operations, and grouping symbols.
An expression formed by repeated multiplication of the same factor is a power.
A power has two parts: an exponent and a base. The exponent represents the
number of times the base is used as a factor. In the power shown below, the
base 7 is used as a factor 3 times.
exponent
73 5 7 p 7 p 7
base
power
You do not usually write the exponent when it is 1. For instance, you can write 81
simply as 8.
EXAMPLE 1
Evaluate powers
a. (25)4 5 (25) p (25) p (25) p (25) 5 625
b. 254 5 2(5 p 5 p 5 p 5) 5 2625
In Example 1, notice how parentheses are used in part (a) to indicate that the
base is 25. In part (b), the base of the power is 5, not 25. An order of operations
helps avoid confusion when evaluating expressions.
KEY CONCEPT
For Your Notebook
Order of Operations
Steps
STEP 1
First, do operations that occur
Example
1 1 72 p (5 2 3)
within grouping symbols.
STEP 2 Next, evaluate powers.
5 1 1 72 p 2
STEP 3 Then, do multiplications and
5 1 1 49 p 2
divisions from left to right.
STEP 4 Finally, do additions and
subtractions from left to right.
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5 1 1 98
5 99
Chapter 1 Equations and Inequalities
10/19/05 2:36:34 PM
VARIABLES A variable is a letter that is used to represent one or more numbers.
An expression involving variables is called an algebraic expression. When you
substitute a number for each variable in an algebraic expression and simplify,
you are evaluating the algebraic expression.
EXAMPLE 2
Evaluate an algebraic expression
Evaluate 24x 2 2 6x 1 11 when x 5 23.
24x2 2 6x 1 11 5 24(23)2 2 6(23) 1 11
"MHFCSB
EXAMPLE 3
Substitute 23 for x.
5 24(9) 2 6(23) 1 11
Evaluate power.
5 236 1 18 1 11
Multiply.
5 27
Add.
at classzone.com
Use a verbal model to solve a problem
CRAFT FAIR You are selling homemade candles at a craft fair for $3 each.
You spend $120 to rent the booth and buy materials for the candles.
• Write an expression that shows your profit from selling c candles.
• Find your profit if you sell 75 candles.
Solution
STEP 1
Write a verbal model. Then write an algebraic expression. Use the fact
that profit is the difference between income and expenses.
Price per candle
(dollars/candle)
Number of candles sold
p
p
3
(candles)
2
c
2
Expenses
(dollars)
120
An expression that shows your profit is 3c 2 120.
STEP 2 Evaluate the expression in Step 1 when c 5 75.
3c 2 120 5 3(75) 2 120
Substitute 75 for c.
5 225 2 120
Multiply.
5 105
Subtract.
c Your profit is $105.
✓
GUIDED PRACTICE
for Examples 1, 2, and 3
Evaluate the expression.
1. 63
2. 226
3. (22) 6
4. 5x(x 2 2) when x 5 6
5. 3y 2 2 4y when y 5 22
6. (z 1 3) 3 when z 5 1
7. WHAT IF? In Example 3, find your profit if you sell 135 candles.
1.2 Evaluate and Simplify Algebraic Expressions
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For Your Notebook
KEY CONCEPT
Terms and Coefficients
variable
terms
In an expression that can be written as a
sum, the parts added together are called
terms.
constant
term
3x2 1 5x 1 (27)
A term that has a variable part is called
a variable term. A term that has no variable
part is called a constant term.
coefficients
When a term is a product of a number and
a power of a variable, the number is called
the coefficient of the power.
SIMPLIFYING An expression is simplified if it contains no grouping symbols and
all like terms are combined. Like terms are terms that have the same variable
parts. (Constant terms are also considered like terms.) The distributive property
allows you to combine like terms by adding coefficients.
EXAMPLE 4
Simplify by combining like terms
a. 8x 1 3x 5 (8 1 3)x
Distributive property
5 11x
Add coefficients.
b. 5p2 1 p 2 2p2 5 (5p2 2 2p2) 1 p
AVOID ERRORS
2
The terms 3p2 and p
are not like terms. They
use the same variable
but different exponents,
so the terms cannot be
combined.
5 3p 1 p
Group like terms.
Combine like terms.
c. 3(y 1 2) 2 4(y 2 7) 5 3y 1 6 2 4y 1 28
Distributive property
5 (3y 2 4y) 1 (6 1 28)
Group like terms.
5 2y 1 34
Combine like terms.
d. 2x 2 3y 2 9x 1 y 5 (2x 2 9x) 1 (23y 1 y)
5 27x 2 2y
Group like terms.
Combine like terms.
IDENTITIES Two algebraic expressions are equivalent expressions if they have
the same value for all values of their variable(s). For instance, in part (a) of
Example 4, the expressions 8x 1 3x and 11x are equivalent. A statement such as
8x 1 3x 5 11x that equates two equivalent expressions is called an identity.
✓
GUIDED PRACTICE
for Example 4
8. Identify the terms, coefficients, like terms, and constant terms in the
expression 2 1 5x 2 6x2 1 7x 2 3. Then simplify the expression.
Simplify the expression.
9. 15m 2 9m
12. 2q2 1 q 2 7q 2 5q2
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10. 2n 2 1 1 6n 1 5
11. 3p3 1 5p2 2 p 3
13. 8(x 2 3) 2 2(x 1 6)
14. 24y 2 x 1 10x 1 y
Chapter 1 Equations and Inequalities
10/19/05 2:36:40 PM
EXAMPLE 5
Simplify a mathematical model
DIGITAL PHOTO PRINTING You send 15 digital images
to a printing service that charges $.80 per print in large
format and $.20 per print in small format. Write and
simplify an expression that represents the total cost if n
of the 15 prints are in large format. Then find the total
cost if 5 of the 15 prints are in large format.
Solution
Write a verbal model. Then write an algebraic expression.
Price of
large print
Number of
large prints
p
(dollars/print)
INTERPRET
EXPRESSIONS
(prints)
p
0.8
The total number of
prints is 15, so if n are
in large format, then
15 2 n are in small
format.
Price of
small print
1
p
Number of
small prints
(dollars/print)
n
p
0.2
1
(prints)
(15 2 n)
An expression for the total cost is 0.8n 1 0.2(15 2 n).
0.8n 1 0.2(15 2 n) 5 0.8n 1 3 2 0.2n
Distributive property
5 (0.8n 2 0.2n) 1 3
Group like terms.
5 0.6n 1 3
Combine like terms.
c When n 5 5, the total cost is 0.6(5) 1 3 5 3 1 3 5 $6.
✓
GUIDED PRACTICE
for Example 5
15. WHAT IF? In Example 5, write and simplify an expression for the total cost if
the price of a large print is $.75 and the price of a small print is $.25.
1.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 21, 29, and 59
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 24, 33, 51, and 59
5 MULTIPLE REPRESENTATIONS
Ex. 61
SKILL PRACTICE
1. VOCABULARY Copy 127 and label the base and the exponent.
2. ★ WRITING Explain what it means for terms to be like terms.
3. ERROR ANALYSIS Describe and correct the error in
234 5 81
evaluating the power shown at the right.
EXAMPLE 1
on p. 10
for Exs. 4–15
EVALUATING POWERS Evaluate the power.
4. 23
5. 34
8. 252
9. 225
12. (23)2
13. (24) 3
6. 43
7. 72
10. 283
11. 2104
14. (22) 8
15. (28)2
1.2 Evaluate and Simplify Algebraic Expressions
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EXAMPLE 2
ORDER OF OPERATIONS Evaluate the expression for the given value of
on p. 11
for Exs. 16–24
the variable.
16. 5d 2 6 when d 5 7
17. 210f 1 15 when f 5 2
18. 6h 4 2 1 h when h 5 4
19. 5j 2 3j p 5 when j 5 10
2
20. (k 1 2) 2 6k when k 5 5
21. 8m 1 (2m 2 9) 3 when m 5 6
22. n3 2 4n 1 10 when n 5 23
23. 2x4 2 4x 3 when x 5 21
"MHFCSB
at classzone.com
24. ★ MULTIPLE CHOICE What is the value of 2x2 2 6x 1 15 when x 5 22?
A 11
B 19
C 35
D 43
EXAMPLE 4
SIMPLIFYING EXPRESSIONS Simplify the expression.
on p. 12
for Exs. 25–33
25. 9x 2 4x 1 5
26. y 2 1 2y 1 3y 2
27. 5z2 2 2z 1 8z2 1 10
28. 10w 2 2 4w 1 3w 2 1 18w
29. 7(m 2 3) 1 4(m 1 5)
30. 10(n2 1 n) 2 6(n2 2 2)
31. 4p2 2 12p 2 9p2 1 3(4p 1 7)
32. 6(q 2 2) 2 2(q2 1 6q)
33. ★ MULTIPLE CHOICE Which terms are like terms?
B 3x2, 4x
A 2x, 2y
C x 2, y 2
D 10x 3, 2x 3
GEOMETRY Write a simplified expression for the perimeter of the figure.
Then evaluate the expression for the given value(s) of the variable(s).
34. a 5 3, b 5 10
35. n 5 2
36. g 5 5, h 5 4
5a 1 b
5a
4n
2b
g 1 2h
n 1 12
EVALUATING EXPRESSIONS Evaluate the expression for the given values of
x and y.
37. 5x 1 6y when x 5 16 and y 5 29
38. 16x 1 11y when x 5 22 and y 5 23
3
HINT
Fraction bars
are grouping
symbols.
39. x 1 5y when x 5 4 and y 5 23
40. (3x)2 2 y 3 when x 5 4 and y 5 5
x2y
41. } when x 5 10 and y 5 8
x1y
x 1 2y
42. } when x 5 23 and y 5 4
4x 2 y
SIMPLIFYING EXPRESSIONS Simplify the expression.
43. 16c 2 10d 1 3d 2 5c
2
2
44. 9j 1 4k 2 2j 2 7k
2
45. 2m 2 5n 1 6n 2 8m
46. p3 1 3q2 2 q 1 3p3
47. 10m2 1 3n 2 8 1 3m2 2 3n 1 3
48. 3y 2 1 5x 2 12x 1 9y 2 2 5
49. 8(s 2 t) 1 16(t 2 s)
50. 3(x2 2 y) 1 9(x2 1 2y)
51. ★ OPEN-ENDED MATH Write an algebraic expression that includes three
coefficients, two like terms, and one constant term. Then simplify the
expression.
14
5 WORKED-OUT SOLUTIONS
on p. WS1
★
5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
GROUPING SYMBOLS Add parentheses to make a true statement.
52. 9 1 12 4 3 2 1 5 15
2
54. 8 1 5 2 6 4 3 5 9
53. 4 1 3 p 5 2 2 5 21
55. 3 p 42 2 23 1 32 5 23
56. CHALLENGE Under what conditions are the expressions (x 1 y)2 and
x2 1 y 2 equal? Are the expressions equivalent? Explain.
PROBLEM SOLVING
EXAMPLE 3
57. MOVIE COSTS In the United States, the average movie ticket price (in dollars)
since 1974 can be modeled by 0.131x 1 1.89 where x is the number of years
since 1974. What values of x should you use to find the ticket prices in 1974,
1984, 1994, and 2004? Find the ticket prices for those years.
on p. 11
for Exs. 57–59
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
58. MILEAGE You start driving a used car when the odometer reads 96,882. After
a typical month of driving, the reading is 97,057. Write an expression for the
reading on the odometer after m months, assuming the amount you drive
each month is the same. Predict the reading after 12 months.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
59. ★ SHORT RESPONSE A student has a debit card with a prepaid amount of
$270 to use for school lunches. The cafeteria charges $4.50 per lunch. Write
an expression for the balance on the card after buying x lunches. Does your
expression make sense for all positive integer values of x? Explain.
EXAMPLE 5
60. CROSS-TRAINING You exercise for 60 minutes, spending w minutes walking
and the rest of the time running. Use the information in the diagram below
to write and simplify an expression for the number of calories burned. Find
the calories burned if you spend 20 minutes walking.
on p. 13
for Exs. 60–62
Walking burns
4 calories per minute.
61.
Running burns
10 calories per minute.
MULTIPLE REPRESENTATIONS A theater has 30 rows of seats with 20 seats
in each row. Tickets for the seats in the n rows closest to the stage cost $45
and tickets for the other rows cost $35.
a. Visual Thinking Make a sketch of the theater seating.
b. Modeling Write a verbal model for the income if all seats are sold.
c. Simplifying Write and simplify an expression for the income.
d. Making a Table Make a table for the income when n 5 5, 10, and 15.
62. COMPUTERS A company offers each of its 80 workers either a desktop
computer that costs $900 or a laptop that costs $1550. Write and simplify an
expression for the cost of all the computers when n workers choose desktop
computers. Find the cost if 65 workers choose desktop computers.
1.2 Evaluate and Simplify Algebraic Expressions
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63. CHALLENGE You want to buy 25 fish for an aquarium. You decide to buy
danios, tetras, and rainbowfish.
danios
$1.50 each
tetras
$2.00 each
rainbowfish
$8.00 each
Write and simplify an expression for the total cost of x danios, y tetras, and
the rest rainbowfish. You buy 8 danios, 10 tetras, and the rest rainbowfish.
What is the total cost?
MIXED REVIEW
PREVIEW
Find the least common denominator (LCD) of the fractions. (p. 979)
Prepare for
Lesson 1.3
in Exs. 64–67.
4, 3
1, }
64. }
}
2 5 10
2, 3
1, }
65. }
}
2 3 4
3, 1, 7
66. }
} }
4 6 8
1, 5
2, }
67. }
}
9 4 6
Identify the property that the statement illustrates. (p. 2)
3 p 7 51
69. }
}
7 3
68. (7 p 8) p 25 5 7 p (8 p 25)
Perform the indicated conversion. (p. 2)
70. 15 meters to centimeters
71. 5000 pounds to tons
72. 100 yards to inches
73. 20 days to minutes
QUIZ for Lessons 1.1–1.2
Graph the numbers on a number line. (p. 2)
7 , 1, 2 4
1. 25, }
}
2
3
2 , 2 Ï}
3. 0, 27.3, 2}
3
5
}
2. 26.2, 5.4, Ï 5 , 22.5
Identify the property that the statement illustrates. (p. 2)
4. 6(4 1 9) 5 6(4) 1 6(9)
5. 25 p 8 5 8 p (25)
6. 17 1 (217) 5 0
Evaluate the expression for the given value of the variable. (p. 10)
7. 10m 1 32 when m 5 25
8. 12 1 (8 2 n) 3 when n 5 5
9. p3 2 3p2 when p 5 22
Simplify the expression. (p. 10)
10. 8x 1 6x 2 2 9x2 2 4x
11. 5(x 1 9) 2 2(4 2 x)
12. 24x 2 6y 1 15y 2 18x
13. CD COSTS CDs are on sale for $8 each and you have a gift card worth $100.
Write an expression for the amount of money left on the gift card after
purchasing n CDs. Evaluate the expression to find the amount of money left
after purchasing 6 CDs. (p. 10)
16
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EXTRA PRACTICE for Lesson 1.2, p. 1010
ONLINE QUIZ at classzone.com
10/19/05 2:36:46 PM
A
Use after Lesson 1.2
classzone.com
Keystrokes
1.2 Evaluate Expressions
QUESTION
How can you use a calculator to evaluate expressions?
You can use a scientific calculator or a graphing calculator to evaluate
expressions. Keystrokes for evaluating several expressions are shown below.
Note that to enter a negative number, you use the
key on a scientific
calculator or the
key (not the
key) on a graphing calculator.
EXAMPLE
Evaluate expressions
EXPRESSION
a. 242 1 6
2
24 1 6
b. (24)2 1 6
2
CALCULATOR
KEYSTROKES
Scientific
4
Graphing
Scientific
RESULT
4
4
6
210
6
210
6
22
(24) 1 6
Graphing
c. (39 4 3) 3
Scientific
39
3
3
2197
(39 4 3)3
Graphing
39
3
3
2197
Scientific
64
5
8
4
6
Graphing
64
5
8
4
6
64 2 5 p 8
d. }
4
64 2 5 p 8
}
4
4
6
22
PRACTICE
Use a calculator to evaluate the expression.
1. 50.2 2 15 4 3
2. 211(20) 2 66
3. 21(28) 1 51
4. (24)4
5. 7(44.5 2 82)
9.2 2 15.9
6. }
219 1 14
Use a calculator to evaluate the expression when x 5 23, y 5 5, and z 5 26.
7. 7z 1 y
10x
10. }
2z 2 3
8. x6
11. (x 1 y)2 1 3z
9. 6y 2 z3
12. (24x 1 9) 4 (y 1 2)
13. ERROR ANALYSIS A student evaluated the expression 7 1 (24) 3 on
a graphing calculator by pressing 7
4
3
. The calculator displayed an error message. Describe and
correct the error.
1.2 Evaluate and Simplify Algebraic Expressions
17