mafs algebra 1 section 01 7x12 - somersetcanyons.com
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Section 1: Expressions Section 1 β Topic 1
Using Expressions to Represent Real-World Situations
The Florida Turnpike is a 320-mile stretch of highway running from Central Florida to South Florida. Drivers who use a SunPass pay discounted toll rates. For 2-axle passenger vehicles with a SunPass, the toll at the Orlando (I-4) Mile Post is $0.53, and the toll at the Okeechobee Plaza exit in West Palm Beach is $1.04. A 2-axle passenger vehicle with a SunPass drives through the Orlando (I-4) Mile Post five times. Determine the total amount of money that will be deducted from this vehicleβs SunPass account. $π. ππ β π $π. ππ Create an algebraic expression to describe the total amount of money that will be deducted from this vehicleβs SunPass account for any given number of drives through the Orlando (I-4) Mile Post. Let π be the number of drives through I-4 Mile Post. $π. ππ β π π. πππ
The same 2-axle passenger vehicle drives through the Okeechobee Plaza exit in West Palm Beach seven times. Determine the total amount of money that will be deducted from its SunPass account. $π. ππ β π $π. ππ Create an algebraic expression to describe the total amount of money that will be deducted from this vehicleβs SunPass account for any given number of drives through the Okeechobee Plaza exit. Let π be the number of drives through Okeechobee. $π. ππ β π π. πππ Write an algebraic expression to describe the combined total amount of money that will be deducted from this vehicleβs SunPass account after it passes through the Orlando (I-4) Mile Post and the Okeechobee Plaza exit any given number of times. Remember that π is the number of drives through I-4 Mile Post and π is the number of drives through Okeechobee π. πππ + π. πππ What is the total amount of money that will be deducted from this vehicleβs SunPass account after it passes through the Orlando (I-4) Mile Post eight times and Okeechobee Plaza exit four times? π. πππ + π. πππ = π. ππ π + π. ππ π = π. ππ The total amount is $π. ππ.
Letβs Practice! 1. Mario and Luigi plan to buy a Nintendo Switch for $299.00.
Nintendo Switch games cost $59.99 each. They plan to purchase one console.
a. Use an algebraic expression to describe how much
they will spend, before sales tax, based on purchasing the console and the number of games.
Let π represent the number of games.
πππ. πππ + ππ. πππ
b. If they purchase one console and three games, how much will they spend before sales tax?
πππ. ππ π + ππ. ππ π = πππ. ππ
c. Mario and Luigi want to purchase some extra controllers for their friends. Each controller costs $29.99. Use an algebraic expression to describe how much they will spend in total, before sales tax, based on purchasing the console, the number of games, and the number of extra controllers. Let π represent the number of controllers
πππ. πππ + ππ. πππ + ππ. πππ
d. What will be the total cost, before sales tax, if Mario and Luigi purchase one console, three games, and two extra controllers? πππ. ππ π + ππ. ππ π + ππ. ππ π = πππ. ππ
Try It! 2. The Thomas family likes to visit Everglades National Park,
the largest tropical wilderness in the U.S., which partially covers the Miami-Dade, Monroe, and Collier counties in Florida. They hold an annual pass that costs $40.00. Each canoe rental costs $26.00 and each bicycle rental costs $15.00. Use an algebraic expression to describe how much they spend in a year based on the annual pass, the number of canoe rentals, and the number of bicycle rentals. Identify the parts of the expression by underlining the coefficient(s), circling the constant(s), and drawing a box around the variable(s). Let π be the number of canoe rentals and π the number of bicycle rentals. Then, ππ + πππ + πππ.
ππ + πππ + πππ
3. Ramiro brought home ten pounds of rice to add to the 24
ounces of rice he had in the pantry. Let π₯ represent the amount of rice Ramiro uses over the next few days. a. Write an algebraic expression to describe the amount
of rice remaining in Ramiroβs pantry. Let π represent number of pounds of rice:
π. π + ππ β π = ππ. π β π
b. Determine if there are any constraints on π₯. π will be integer greater than zero but less than or equal ππ. π (amount of rice available to use)
When defining variables, choose variables that make sense to you, such as β for hours and π for days.
BEAT THE TEST!
1. JosΓ© is going to have the exterior of his home painted. He will choose between Krystal Klean Painting and Elegance Home Painting. Krystal Klean Painting charges$175.00 to come out and evaluate the house plus $7.00 for every additional 30 minutes of work. Elegance Home Painting charges $23.00 per hour of work. Let β represent the number of hours for which JosΓ© hires a painter. Which of the following statements are true? Select all that apply.
Β¨ The expression 14β represents the total charge for
Krystal Klean Painting. Γ½ The expression 23β represents the total charge for
Elegance Home Painting. Β¨ The expression 175 + 14β + 23β represents the total
amount JosΓ© will spend for the painters to paint the exterior of his home.
Γ½ If JosΓ© hires the painters for 10 hours, then Elegance Home Painting will be cheaper.
Γ½ If JosΓ© hires the painters for 20hours, then Krystal Klean Painting will be cheaper.
2. During harvesting season at Florida Blue Farms, hand pickers collect about 200 pounds of blueberries per day with a 95% pack-out rate. The blueberry harvester machine collects about 22,000 pounds of blueberries per day with a 90% pack-out rate. The pack-out rate is the percentage of collected blueberries that can be packaged to be sold, based on Florida Blue Farms' quality standards. Let β represent the number of days the hand pickers work and π represent the number of days the harvester machine is used. Which of the following algebraic expressions can be used to estimate the amount of collected blueberries that are packed at the Florida Blue Farms for fresh consumption this season?
A 0.95β + 200 0.90π + 22000 B π. ππ ππππ + π. ππ(ππππππ) C 200.95β + 22000.90π D 200β + 0.95 22000π + 0.90
Answer is B.
Want to learn more about how Florida Blue Farms uses algebra to harvest blueberries? Visit the Student Area in Algebra Nation to see how people use algebra in the real world! You can find the video in the βMath in the Real World: Algebra at Workβ folder.
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Section 1 β Topic 2 Understanding Polynomial Expressions
A term is a constant, variable, or multiplicative combination of the two. Consider 3π₯K + 2π¦ β 4π§ + 5. How many terms do you see? π List each term. πππ ππ βππ π This is an example of a polynomial expression. A polynomial can be one term or the sum of several terms. There are many different types of polynomials. A monarchy has one leader. How many terms do you think a monomial has? π A bicycle has two wheels. How many terms do you think a binomial has? π A triceratops has three horns. How many terms do you think a trinomial has? π
Letβs recap:
Type of Polynomial Number of Terms Example
Monomial π πππ
Binomial π ππ + π
Trinomial π πππ + ππ + ππ
Polynomial π or more ππ+ ππ +πππ + π
Some important facts:
Γ The degree of a monomial is the sum of the ____________ of the variables.
Γ The degree of a polynomial is the degree of the
monomial term with the ____________ degree. Sometimes, you will be asked to write polynomials in standard form.
Γ Write the monomial terms in ________________ _________ order.
Γ The leading term of a polynomial is the term with the
________________ _____________. Γ The leading coefficient is the coefficient of the
_____________ _________.
exponents
highest
descending
degree
highest degree
leading degree
Letβs Practice! 1. Are the following expressions polynomials? If so, name the
type of polynomial and state the degree. If not, justify your reasoning.
a. 8π₯Kπ¦S
Yes, monomial, degree 5
b. KTU
SV
No, it is the quotient of variables
c. SKπ₯W β 5π₯S + 9π₯X
Yes, trinomial, degree 7
d. 10πZπK + 17ππSπ β 5πX Yes, trinomial, degree 8
e. 2π + 3π^_ + 8πKπ No, it has the quotient of a constant and variable.
Try It! 2. Are the following expressions polynomials?
a. _
Kπ + 2πK
l polynomial o not a polynomial
b. 34
l polynomial o not a polynomial
c.
`aaU
o polynomial l not a polynomial
d. 2ππ + π W
l polynomial o not a polynomial
e. π₯π¦K + 3π₯ β 4π¦^_
o polynomial l not a polynomial
3. Consider the polynomial 3π₯W β 5π₯S + 9π₯X.
a. Write the polynomial in standard form.
πππ + πππ β πππ
b. What is the degree of the polynomial?
π
c. How many terms are in the polynomial?
π
d. What is the leading term?
πππ
e. What is the leading coefficient?
π
BEAT THE TEST! 1. Match the polynomial in the left column with its
descriptive feature in the right column.
A. π₯S + 4π₯K β 5π₯ + 9 V I. Fifth degree polynomial
B. 5πKπS I II. Constant term of β2
C. 3π₯W β 9π₯S + 4π₯d VII III. Seventh degree polynomial
D. 7πZπK + 18ππSπ β 9πX VI IV. Leading coefficient of 3
E. π₯e β 9π₯S + 2π₯X III V. Four terms
F. 3π₯S + 7π₯K β 11 IV VI. Eighth degree polynomial
G. π₯K β 2 II VII. Equivalent to 4π₯d + 3π₯W β 9π₯S
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Section 1 β Topic 3 Algebraic Expressions Using the Distributive Property
Recall the distributive property.
Γ If π and π are real numbers, then π π + π = π β _____ + π β ______.
One way to visualize the distributive property is to use models. Consider (π + 3)(π + 2).
Now, use the distributive property to write an equivalent expression for (π + 3)(π + 2). π β π + π β π + π β π + π β π = ππ + ππ + ππ + π = ππ + ππ + π
3! "
! !" "!
# #! $
!" + &! + $
ππ
Letβs Practice! 1. Write an equivalent expression for 3(π₯ + 2π¦ β 7π§) by
modeling and then by using the distributive property.
π π + ππ β ππ = π β π + π β ππ β π β ππ = ππ + ππ β πππ
2. Write an equivalent expression for (π₯ β 3)(π₯ β 2) by
modeling and then by using the distributive property.
π β π π β π = π β π β π β π β π β π β π β βπ = ππ β ππ + π
!
" #$ β&'
!" ($ β#)'
3! β#
! !# β#!
β$ β$! %
!# β &!+ %
Try It! 3. Use the distributive property or modeling to write an
equivalent expression for π + 5 π β 3 .
π+ π πβ π = π βπ+π β βπ + π β π + π β βπ =
ππ + ππβ ππ
3! β#
! !$ β#!
% %! β&%
!$ + $!β &%
BEAT THE TEST! 1. Students were asked to use the distributive property to
write an equivalent expression for the expression π₯ β 5 π₯ β 2 . Their work is shown below. Identify the
student with the correct work. For the answers that are incorrect, explain where the students made mistakes.
Student 1
Incorrect. Student did not correctly distribute. Student 2
Correct. Student 3
Incorrect. βπ βπ = ππ, not βππ.
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Section 1 β Topic 4 Algebraic Expressions Using the
Commutative and Associative Properties What is 5 + 2? What is 2 + 5? π π Does it matter which number comes first? No What is 9 β 2? What is 2 β 9? ππ ππ Does it matter which number comes first? No This is the commutative property.
Γ The order of the numbers can be _____________ without affecting the _________ or ______________.
Γ If π and π are real numbers, then π + π = ___________ and
π β π = ____________. Does the commutative property hold true for division or subtraction? If so, give an example. If not, give a counterexample. No, π β π β π β π and ππ
πβ π
ππ
changed
sum product
π + π
π β π
Letβs look at some other operations and how they affect numbers. Consider 2 + 4 + 6. What happens if you put parentheses around any two adjacent numbers? How does it change the sum? π + π + π π + (π + π) It does not change the sum. Consider 3 β 6 β 4. What happens if you put parentheses around any two adjacent numbers? How does it change the product? (π β π) β π π β (π β π) It does not change the product. This is the associative property.
Γ The ____________ of the numbers does not change. Γ The grouping of the numbers can change and does
not affect the ___________ or ______________.
Γ If π, π, and π are real numbers, then π + π + π = ____________________ and
ππ π = ______________. Does the associative property hold true for division or subtraction? If not, give a counterexample. No, ππ β π β π β ππ β π β π and ππ Γ· (π Γ· π) β (ππ Γ· π) Γ· π.
order
sum product
π + (π + π)π(ππ)
Letβs Practice! 1. Name the property (or properties) used to write the
equivalent expression.
a. [5 + β3 ] + 2 = 5 + [(β3) + 2] Associative property of addition
b. (8 β 4) β 6 = 6 β (8 β 4) Commutative property of multiplication
c. π + π’ + π‘ = π‘ + π’ + π
Commutative property of addition
When identifying properties, look closely at each piece of the problem. The changes can be very subtle.
Try It! 2. Identify the property (or properties) used to find the
equivalent expression.
a. (11 + 4) + 5 = (5 + 11) + 4Commutative property of addition Associative property of addition
b. π£ β (π¦ β π) = (π£ β π¦) β π Associative property of multiplication
c. (8 + 1) + 6 = 8 + (1 + 6)
Associative property of addition
d. 9Γ13 Γ14 = 13Γ9 Γ14
Commutative property of multiplication 3. The following proof shows (3π₯)(2π¦) is equivalent to 6π₯π¦. Fill
in each blank with either βcommutative propertyβ or βassociative propertyβ to indicate the property being used.
3π₯ (2π¦) = 3 π₯ β 2 π¦ _________________________________
= 3 2 β π₯ π¦ _________________________________
= (3 β 2)(π₯ β π¦) _________________________________
= 6π₯π¦
Associative Prop. Of Multiplication
Associative Prop. Of Multiplication
Commutative Prop. Of Multiplication
BEAT THE TEST! 1. Fordson High School hosted a math competition. A
student completed the following proof and the question was marked incorrect. Identify and correct the studentβs mistake(s).
Step 4 is incorrect. In Associative Property of Addition, the order of the numbers does not change. In step 4, the student changed the order of the terms in the sum. The correct answer is Commutative Property of Addition.
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Section 1 β Topic 5 Properties of Exponents
Letβs review the properties of exponents. 2W = π β π β π β π = ππ 2S = π β π β π = π 2K = π β π = π 2_ = π What pattern do you notice? We are dividing by π each time. Continuing the pattern, what does the following term equal? 2q = ππ Γ· π = π Γ· π = π
Γ This is the zero exponent property: πq = ______. Continuing the pattern, what do the following terms equal?
2^_ = ππ Γ· π = π Γ· π =ππ =
πππ
2^K = π^π Γ· π =ππ Γ· π =
ππ βππ =
πππ
Γ This is the negative exponent property: π^r = ______ and
_Tst
= ______.
π
πππ
ππ
Letβs explore multiplying terms with exponents and the same base. 2S β 2W = π β π β π β π β π β π β π = ππ
2e β 2^S = π β π β π β π β π βπ
π β π β π = π β π = ππ π₯S β π₯K = π β π β π β π β π = ππ
Γ This is the product property: πu β πr = ______. Letβs explore dividing terms with exponents and the same base. Wv
Ww= πβπβπβπβπ
πβπβπ= ππ
`x
`y= πβπβπβπβπβπβπ
πβπβπβπβπβπβπβπ= π
π= π^π
Γ This is the quotient property: TzTt
= ______.
Letβs explore raising powers to an exponent. (5S)K = ππ β ππ = π π{π = ππ π¦W S = ππ β ππ β ππ = π π{π{π = πππ
Γ This is the power of a power property: (πu)r = ______.
ππ{π
ππ^π
πππ
Letβs explore raising a product to an exponent. 2 β 3 K = π β π β π β π = π β π β π β π = ππ β ππ
4 β π₯ S = π β π β π β π β π β π = π β π β π β π β π β π = ππ β ππ
Γ This is the power of a product property:(ππ)r = ________. Letβs explore a quotient raised to an exponent. KqS
K= ππ
πβ πππ= ππβππ
πβπ= πππ
ππ
Za
S= π
πβ ππβ ππ= ππ
ππ
Γ This is the power of a quotient property: ππ
π= ________.
ππππ
ππ
ππ
Letβs Practice! 1. Determine if the following equations are true or false.
Justify your answers.
a. 3S β 3W = (39)(32)
True
ππ β ππ = π π{π = ππ and (ππ)
(ππ)= π(π^π) = ππ
b. (5 β 4K)S = 5W β 5q β W|
es}
^_
False
(π β ππ)π = ππ β ππ π = ππ β ππ
ππ β ππ β ππ
πβπ
^π
= ππ β π β πβπ
ππ= ππβπ
ππ= ππ
ππ
Try It! 2. Use the properties of exponents to match each of the
following expressions with its equivalent expression.
A) XK
W II I. 7S β 2Z
B) (7 β 2K)S I II. X~
K~
C) (7K)(7K) IV III. K~
X~
D) 7K 7 q V IV. 7W
E) XK
^W III V. 7K
F) (X|)(Xw)
VI VI. 7S
BEAT THE TEST! 1. Crosby and Adam are working with exponents.
Part A: Crosby claims that 3S β 3K = 3e. Adam argues that
3S β 3K = 3Z. Which one of them is correct? Use the properties of exponents to justify your answer.
Crosby is correct. ππ β ππ = ππ{π = ππ. Adam multiplied the exponents instead of adding.
Part B: Crosby claims that Sy
SU= 34. Adam argues that
Sy
SU= 36. Which one of them is correct? Use the
properties of exponents to justify your answer.
Adam is correct. ππ
ππ= ππ^π = ππ. Crosby divided the
exponents instead of subtracting.
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Section 1 β Topic 6 Radical Expressions and Expressions with Rational
Exponents Exponents are not always in the form of integers. Sometimes you will see them expressed as rational numbers. Consider the following expressions with rational exponents. Use the property of exponents to rewrite them as radical expressions.
9}U = ππ
ππ = π πβππ = π 8
}w = ππ
ππ = π πβππ = π
Do you notice a pattern? If so, what pattern did you notice? πππ = π = π and π
ππ = π = ππ
Consider the following expression with rational exponents. Use the pattern above and the property of exponents to rewrite them as radical expressions.
2Uw = ππ
ππ = πππ 5
wU = ππ
ππ = ππ
or ππππ= ππ π
or ππππ= π
π
For exponents that are rational numbers, such as TV, we
haveπ₯οΏ½οΏ½ = ______________ = ______________ .
βπππ οΏ½βππ οΏ½π
Letβs Practice! 1. Use the rational exponent property to write an equivalent
expression for each of the following radical expressions. a. π₯ + 2
π + πππ
b. π₯ β 5w + 2
π β πππ + π
2. Use the rational exponent property to write each of the
following expressions as integers. a. 9
}U
π = π
b. 16}U
ππ = π
c. 8}w
ππ = π
d. 8Uw
ππ π= ππ = π
e. 125Uw
ππππ π= ππ = ππ
f. 16w~
πππ π= ππ = π
Try It! 3. Use the rational exponent property to write an equivalent
expression for each of the following radical expressions. a. π¦
πππ
b. π¦ + 6v β 3
π + πππ β π
4. Use the rational exponent property to write each of the
following expressions as integers.
a. 49}U ππ = π
b. 27
}w
πππ = π c. 216
Uw
ππππ π= ππ = ππ
BEAT THE TEST! 1. Match each of the following to its equivalent expression.
A. 2}w VI 1. π
}U β 3
B. π β 3 III 2. (3π)}U
C. 2Uw V 3. (π β 3)
}U
D. π β 3 I 4. 2
E. 2}U IV 5. 4w
F. 3π II 6. 2w
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Section 1 β Topic 7 Adding Expressions with Radicals and Rational Exponents Letβs explore operations with radical expressions and expressions with rational exponents. For each expression, label approximately where the answer would be found on the number line. 5 + 2
Since the radicands are not the same, we cannot add the radicals.
5_K + 2
_K
Since the bases are not the same, we cannot add exponents.
5 + 5 π π
5}U + 5
}U
π β π
ππ
2 3 β 8 3 βπ π
2 β 3}U β 8 β 3
}U
βπ β π
ππ
To add radicals, the radicand of both radicals must be the same. To add expressions with rational exponents, the base and the exponent must be the same. In both cases, you add only the coefficients.
123456 123456
123456 123456
β12β11β10 β 9 β 8 β 7 β12β11β10 β 9 β 8 β 7
Letβs Practice! 1. Perform the following operations.
a. 12 + 3 = π β π + π = π π + π = π π
b. 12}U + 3
}U
= π β πππ + π
ππ
= πππ β π
ππ + π
ππ
= π β πππ + π
ππ
= π β πππ
c. 72 + 15 + 18 = ππ β π + ππ + π β π = π π + ππ + π π = π π + ππ
d. 72}U + 15
}U + 18
}U
= ππ β πππ + ππ
ππ + π β π
ππ
= ππππ β π
ππ + ππ
ππ + π
ππ β π
ππ
= π β πππ + ππ
ππ + π β π
ππ
= π β πππ + ππ
ππ
e. 32 + 16w = ππ β π + π β ππ = π π + π ππ
f. 32}U + 16
}w
= ππ β πππ + π β π
ππ
= ππππ β π
ππ + π
ππ β π
ππ
= π β πππ + π β π
ππ
For radicals and expressions with rational exponents, always look for factors that are perfect squares when taking the square root (or perfect cubes when taking the cube root).
Try It! 2. Perform the following operations.
o 6 + 3 6 = π π
o 6}U + 3 β 6
}U
= π β πππ
o 50 + 18 + 10 = ππ β π + π β π + ππ = π π + π π + ππ = π π + ππ
o 50}U + 18
}U + 10
}U
= ππ β πππ + π β π
ππ + ππ
ππ
= ππππ β π
ππ + π
ππ β π
ππ + ππ
ππ
= π β πππ + π β π
ππ + ππ
ππ
= π β πππ + ππ
ππ
o 2w + 8w + 16w = ππ + π + π β ππ = ππ + π + π ππ = π ππ + π
o 2}w + 8
}w + 16
}w
= πππ + π + π β π
ππ
= πππ + π + π
ππ β π
ππ
= πππ + π + π β π
ππ
= π β πππ + π
BEAT THE TEST! 1. Which of the following expressions are equivalent to 3 2?
Select all that apply. o 3
}U + 2
}U
Γ½ 8}U + 2
}U
Γ½ 3 β 2}U
Γ½ 18 o 2 18 Γ½ 8 + 2
2. Miguel completed a proof to show that 27 + 3 = 4 β 3
}U :
27 + 3 = 27
}U + 3
}U
= _________ = 3 β 3
}U + 3
}U
= 4 β 3_K
Part A: Which expression can be placed in the blank to
correctly complete Miguelβs proof?
A 3}U 9
}U + 3
}U
B π β πππ + π
ππ
C 9}U + 3
}U + 3
}U
D 9}U + 3
}U
Answer: B
Part B: Label and place 4 β 3}U on the number line below.
4 β 3_K = 6.9282
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ππππππ
Section 1 β Topic 8 More Operations with Radicals and Rational Exponents
Letβs explore multiplying and dividing expressions with radicals and rational exponents. 10 β 2 = ππ β π = ππ = π β π = π π
10_K β 2
_K
= π β πππ β π
ππ
= (πππ β π
ππ) β π
ππ
= (πππ β π
ππ) β π
ππ
= πππ { π
π β πππ
= π β πππ
2 β 2w We cannot multiply the radicals since the roots are not the same.
2_K β 2
_S
= πππ { π
π
= πππ
102
=πππ
= π
10_K
2_K
=π β π
ππ
πππ
=πππ β π
ππ
πππ
= πππ
Letβs Practice! 1. Use the properties of exponents to perform the following
operations.
a. π₯}w
}U = π
ππ β ππ = π
ππ
b. 7
S= ππ = ππ β π = π π
c. π}Uπ
Uv β π
Uwπ
}U = π
ππ β π
ππ β π
ππ β π
ππ = π
ππ { π
π β πππ { π
π
= πππ β π
πππ = π π{ππ β π
πππ = π β ππ β ππππ
d. οΏ½~
οΏ½= π
ππ
πππ= π
π
πβπ
π = πβπ
π =π
πππ= π
ππ
The properties of exponents also apply to expressions with rational exponents.
Try It! 2. Use the properties of exponents to perform the following
operations.
a. (πqπK)}v = ππβ ππ β ππβ
ππ = ππ β π
ππ = π β π
ππ = π
ππ
b. 8 β 3wUw = π
ππ β π
ππ
ππ= π
ππ β ππ β π
ππ β ππ = π
ππ β π
ππ
= π β πππ = π β πππ = π β ππ
c. 4~ β 4w = π
ππ β π
ππ = π
ππ { π
π = ππππ = ππππ
d. 3 β 27|}U = π β ππ
ππ
ππ= π β π
ππ
ππ= π
ππ β π
ππ = π
ππ = πππ
BEAT THE TEST! 1. Which of the following expressions are equivalent to 2
}U?
Select all that apply. o 4w o 8w Γ½ 4~ Γ½ 8| o 16|
4w = 4_S = (2K)
_S = 2
KS
8w = 8_S = (2S)
_S = 2
4~ = 4_W = (2K)
_W = 2
_K
8| = 8_Z = (2S)
_Z = 2
_K
16| = 16_Z = (2W)
_Z = 2
KS
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Section 1 β Topic 9 Operations with Rational and Irrational Numbers
Letβs review rational and irrational numbers.
Γ Numbers that can be represented as TV
,where πand π are integers and π β 0, are called ___________________ numbers.
Γ Numbers that cannot be represented in this form are
called ________________ numbers.
o Radicals that cannot be rewritten as integers are examples of such numbers.
Determine whether the following numbers are rational or irrational.
Rational Irrational
π l β π β l π β l πππ l β
π. ππ l β πππ l β
2.23606β¦ β l βππ l β
rational
irrational
Given two rational numbers, πand π, prove that the sum of π and π is rational. The sum is always rational. Examples: π + π = π, π
π+ π
π= π
π
Given two rational numbers, πand π, what can be said about the product of πand π? The product is always rational. Examples: π β βπ = ππ, π. π β π. π = π. ππ
Given a rational number, π, and an irrational number, π, prove that the sum of πand π is irrational. The sum is always irrational. Examples: π + ππ = π + π π, π + ππ = π + ππ Given a non-zero rational number, π, and an irrational number, π, what can be said about the product of πand π? The product is always irrational. Examples: π β π, ππ
Letβs Practice! 1. Consider the following expression.
2 + 3
The above expression represents the
of a(n) and a(n) and is equivalent to a(n)
l sum o product
l rational number o irrational number
o rational number l irrational number
o rational number l irrational number
Try It! 2. MarΓa and her 6best friends are applying to colleges. They
find that Bard College accepts _S of its applicants. MarΓa
and her friends write the expression below to represent how many of them would likely be accepted.
7 β13
The above expression represents the
of a(n) and a(n)
and is equivalent to a(n)
l rational number. o irrational number.
l rational number o irrational number
l rational number o irrational number
o sum lproduct
BEAT THE TEST! 1. Let π and π be non-zero rational numbers and π and π be
irrational numbers. Consider the operations below and choose whether the result can be rational, irrational, or both.
Rational Irrational
π + π Γ½ Β¨ π β π Β¨ Γ½ π β π Γ½ Β¨ ππ Γ½ Γ½
π + π Γ½ Γ½ π β π β π Β¨ Γ½
2. Consider π₯ β π¦ = π§. If π§is an irrational number, what can be
said about π₯ and π¦?
At least one of the variables MUST BE irrational.
Test Yourself! Practice Tool
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