Download File - Cache County School District

Cache County School District 2013-2014
Secondary II
Utah Integrated
Mathematics Core
Student Edition - Honors
Unit 1:
Expanding Our
Number System
Secondary II Unit 1 – Expanding Our Number System: Table of
Contents
Homework Help ....................................................................................................................
Section 1.1 – Reviewing the Real Number System, Teacher Notes ......................................
Notes, Task, Assignment ............................................................................................
Section 1.2 – Rational vs. Irrational Numbers Task, Teacher Notes ...................................
Notes, Assignment .......................................................................................................
Section 1.3 – Reviewing Integer Exponent Properties, Teacher Notes ................................
Notes, Assignment .......................................................................................................
Section 1.4 – Extending to Rational Exponents, Teacher Notes ..........................................
Notes, Assignment .......................................................................................................
Section 1.5 –Perfect Roots with Variables, Teacher Notes ..................................................
Notes, Assignment .......................................................................................................
Section 1.6 – Negative Roots and the Quotient Property, Teacher Notes ...........................
Notes, Assignment .......................................................................................................
Section 1.7 – Radicals and Rational Exponents Task, Teacher Notes .................................
Notes, Assignment .......................................................................................................
Section 1.8 – Adding and Subtracting Radicals, Teacher Notes ..........................................
Notes, Assignment .......................................................................................................
Section 1.9 – Multiplying Radicals, Teacher Notes .............................................................
Notes, Assignment .......................................................................................................
Additional Practice Multiply Radicals Worksheet……………………………………...
Section 1.10 – Quotients and Rationalizing Denominators, Notes ......................................
Assignment ..................................................................................................................
Section 1.11 – Rational Exponents Task, .............................................................................
Assignment ..................................................................................................................
Section 1.12 HONORS– Rationalizing Expressions, Teacher Notes ..................................
Notes, Assignment .......................................................................................................
Unit 1 Review (not a test review)…………………………………………………………
Targeting for Success: Exponents Classroom Activity…………………………………
Secondary II Unit 1 – Expanding Our Number System:
Homework Helps
Section 1.1
Video
http://tinyurl.com/n2ma26v
http://goo.gl/qAKMd
http://goo.gl/isKJx
Section 1.2
video
resources
http://goo.gl/MS5vf
http://goo.gl/uaapn
http://goo.gl/IXpZq
Section 1.3
http://goo.gl/H6U9G
http://goo.gl/7prDD
http://goo.gl/DMXb4
Section 1.4
http://goo.gl/LniWi
http://goo.gl/MFBEo
http://goo.gl/bfP8v
Section 1.5
http://goo.gl/p6DBI
http://goo.gl/6STPL
http://goo.gl/4OnBE
http://goo.gl/3YZc8
http://goo.gl/VxI2f
http://goo.gl/z7oXt
http://goo.gl/7laK3
http://goo.gl/4vByH
http://goo.gl/37T5u
http://goo.gl/x6hth
http://goo.gl/MUl9y
Section 1.6
http://goo.gl/g0v7o
Section 1.7
http://goo.gl/murTP
Section 1.8
http://goo.gl/xnDW3
Section 1.9
http://goo.gl/clHt5
Section 1.10
http://goo.gl/Z8Aua
http://goo.gl/176DY
http://goo.gl/UXlrp
http://goo.gl/RwvXE
http://goo.gl/Whwxd
http://goo.gl/176DY
http://goo.gl/MexH5
Section 1.11
http://goo.gl/4qfk1
Section 1.12
http://goo.gl/UOsjH
Unit 1 Lesson 1 – Reviewing the Real Number System
Notes 1.1
Real Numbers
Rational Numbers: consists of all numbers that can be written as the ratio of
𝟏
̅̅̅̅
two integers. ex: 2, 0.125, , 𝟎. 𝟗𝟏
𝟑
Integers: consists of the whole numbers and their opposites.
(… − 𝟑, −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, 𝟑 … )
Whole Numbers:
Natural Numbers:
(counting numbers)
include zero, and the
natural numbers.
Irrational Numbers: consist of all numbers that cannot be written as the ratio of two
4
integers (non-repeating, non-terminating decimals). Ex: 𝜋, √3, √5
Unit 1 Lesson 1 – Reviewing the Real Number System
Task 1.1
Name______________________________
Date_________ Hour_______
Part 1
Mark an X for each category that applies.
Number
1
-6
2
62%
3
0
4
/2
5
2.7
6
2/5
7
7
8
25
9
1
10
½
11
-3
12
4.75
13
3/4
14
1%
15
9/2
16
4½
17
4.5
18
52
19
23/3
Real Rational Irrational Integer Whole Natural
20
2/3
21
12 5/8
22 1,000,000
23
-4982
24
17.1
25
-17.1
26
3
27
9
28
3/1
29
3.0
30
-15/3
Part 2 Give a number that would satisfy the given rules.
a. A number that is: real, rational, whole, an integer, and natural.
b.
c.
A number that is: real and rational and not whole.
A number that is: real and irrational.
Part 3
Given examples of rational numbers that fit between:
a.
0.56 and -0.65
b.
−5.76 and −5.77
c.
3.64 and 3.46
Part 4
a.
Is the set of odd numbers closed under addition? Why or why not?
b.
Is the set of whole numbers closed under subtraction? Why or why not?
Unit 1 Lesson 1 – Reviewing the Number System
Ready, Set, Go! - Assignment 1.1
http://goo.gl/qAKMd
Name______________________________
Date_________ Hour_______
Ready
Classify each number in as many ways as possible.
1. -23
2. -5.1
4.
2
5.
2
3
3. −√16
9
6.
5
7
7. 0.85
8. 
10. 0
11. - 
9.
3
25
12. 5.010010001….
Set
Graph each pair of numbers on a number line.
13.
-3 and -2.5
14. -1.5 and -4
15.
13
and 7
2
3
and 2
8
17. 3.6 and -4
18.
7 and 3
16. 4
Go!
Evaluate each expression by using the order or operations.
19. 3  22  3
20. 6  3  2
21. 22 (2  3)  5
22. 6  (3  1)  5
23. 3  52  16
24. 5(2  3)2
25. (22  1)  4  2
26. 2(31)  (3  1)
27. 3  4  2(41)
28.
82
 (2  1)
3
29. 2  4 
14
52
30.
Is the set of all positive rational numbers closed under subtraction? Why or why not?
31.
Is the set of all integers closed under addition? Why or why not?
Unit 1 Lesson 2 – Rational vs. Irrational Numbers
Task 1.2
Name____________________________
Date________ Hour________
Choose any ten rational numbers and ten irrational numbers and write them in the space below.
Rationals
Irrationals
Classify the following situations by using the numbers you chose above and writing in the workspace
below:



The sum of two rational numbers is ___________________.
The product of two rational numbers is ___________________.
The sum of a rational number and an irrational number is ___________________.
Workspace:
Summary of sums and products of rational and irrational numbers:
Complete the table. We will be discussing as a class. Be sure to explain your answers.
STATEMENT
The sum of a rational number
and an irrational number is
irrational.
The sum of two rational
numbers is rational.
The product of a rational
number and an irrational
number is irrational.
The sum of two irrational
numbers is irrational.
The product of two rational
numbers is irrational.
The product of two irrational
numbers is irrational.
ALWAYS, SOMETIMES,
OR NEVER TRUE
Justification
Unit 1 Lesson 2 – Rational vs. Irrational Numbers
Ready, Set, Go! - Assignment 1.2
Video
http://goo.gl/MS5vf
Name______________________________
Date_________ Hour_______
Ready
Evaluate/estimate each expression then classify the result as rational or irrational. Round your answers
to the nearest two decimal places.
𝟏. 1 − 𝜋
2. √2 ∙ √8
3. √6 + √3
4.
𝜋
2
Classify each statement as always true, sometimes true, or never true.
5. Between two rational numbers there is an irrational number.
6. An simplified expression containing both √6 and 𝜋 is irrational.
Set:
Classify each statement as always true, sometimes true, or never true.
7. Between two irrational numbers there is an irrational number.
8. The product of a rational number and an irrational number is irrational.
9. The product of two irrational numbers is irrational.
Go!
13.
a. Write three rational numbers.
b. Explain what a rational number is in your own words.
14.
a. Write three irrational numbers.
b. Explain what an irrational number is in your own words.
Unit 1 Lesson 3 Reviewing Integer Exponent Properties
Notes 1.3
Vocabulary
Exponent –
Base –
Rational Exponent –
Properties of Exponents
(all bases are non-zero)
𝑥𝑎 ∙ 𝑥𝑏 =
𝑥𝑎
=
𝑥𝑏
𝑥0 =
𝑥 −𝑎 =
(𝑥𝑦)𝑛 =
(𝑥 𝑚 )𝑛 =
𝑥 𝑚
( ) =
𝑦
Examples
Write an equivalent form of each expression. Your answer should contain no negative variables
and each variable should only be represented once.
1.
52 ∙ 53
2.
𝑡3 ∙ 𝑡5 ∙ 𝑡
3.
(𝑎3 )6
5a.
−20
5c.
(−2𝑥)0
6.
𝑞 −2
10.
𝑎5 𝑏 7 𝑐 −2
𝑎−1 𝑏 2 𝑐
5𝑎2 𝑏 5
5b.
5d.
7.
3
( 𝑏2 𝑐6 )
4.
2(−2)0
−2𝑥 0
−2𝑎𝑥 −3
11.
8.
(3𝑥 −1 𝑦 3 )
(3𝑎𝑏)−1
−2
(3𝑥𝑦 −1 )3
∙ (9𝑥 −9 𝑦 5 )
Unit 1 Lesson 3 – Reviewing Integer Exponent Properties
Ready, Set, Go! - Assignment 1.3
http://goo.gl/H6U9G
Name______________________________
Date_________ Hour_______
Ready
True or False, IF FALSE, STATE THE CORRECT ANSWER:
1.
33 = 9
2.
−23 = 8
4.
23 ∙ 33 = 66
3.
(𝑥 7 )3 = 𝑥10
5.
Explain how the expressions (−2)2 𝑎𝑛𝑑 −22 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡:
6.
Explain how the expressions (5𝑥)3 𝑎𝑛𝑑 5𝑥 3 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡:
Identify the base and the exponent for each exponential expression. Evaluate each expression.
7.
−27
8.
(−3)5
9.
(−2𝑥)4
10.
Explain why the product rule does not apply to the expression 52 + 53 . Next, explain the correct
way to simplify the expression.
11.
Explain why the product rule does not apply to the expression 32 ∙ 43 . Then, simplify the
expression.
Set
Write an equivalent form of each expression. Your answer should contain no negative variables and
each variable should only be represented once.
12.
14.
16.
17.
(𝑎4 𝑏 5 𝑐 2 )4 (𝑎𝑏 3 𝑐)2
13.
3𝑎5 𝑏3 𝑐 3
15.
9𝑎3 𝑏7 𝑐
1
𝑡2
𝑡5
( 𝑡 ) ( 5 ) (𝑡 3)
8𝑎3 𝑏2
4
(16𝑎2𝑏3)
3𝑎(5𝑎2 𝑏)(6𝑎𝑏 3 )
36
3−2
18.
10𝑥 −6
19.
−2𝑥 2
2−3 𝑤 5
𝑤 3 𝑤 −7
Go!
Write an equivalent form of each expression. Your answer should contain no negative variables and
each variable should only be represented once.
20.
(3𝑥)−3 (𝑥 2 𝑦 −2 )−2
21.
2𝑟 −3 𝑡 −1
10𝑟 5 𝑡 2
22.
24.
26.
5𝑎 ∙ 𝑎2 + 3𝑎 −2 ∙ 𝑎5
(2𝑥 −2 𝑦)−3
(2𝑥𝑦 −1 )2
(3𝑥 −1 𝑦 3 )−2
(3𝑥𝑦 −1 )3
∙ (2𝑥 2 𝑦 −7 )
∙ (9𝑥 −9 𝑦 5 )
3
−2
23.
(− 4𝑥 3 )
25.
( 3𝑥𝑦3 )
2𝑥 3 𝑦 2
−1
Unit 1 Lesson 4 - Extending to Rational Exponents
Notes 1.4
Properties of Rational Exponents
(all bases are non-zero)
𝑥
𝑝
⁄𝑞
∙𝑥
𝑟⁄
𝑠
𝑝
𝑥𝑞
𝑟
=
𝑥𝑠
𝑥
−𝑝
⁄𝑞
=
𝑝
(𝑥𝑦)𝑞 =
𝑟
𝑝 𝑠
(𝑥 𝑞 )
𝑝
=
𝑥 𝑞
( ) =
𝑦
=
Examples
Examples: Write an equivalent form of each expression. Your answer should contain no negative
variables and each variable should only be represented once.
1
1
1. 3 ∙ 35
3.
2
2. 43 ∙ 43
1
5
1
4. 52 ∙ 54
2
53
2
1
123
5. (
1
43
)
6.
1
3
1
2
1
−
𝑘 3
2
7. (8 ∙ 5 )
9. 5𝑦
1
1
2
−
8. 𝑦
−
2
3
10
10.
1
104
a. 5
1⁄
2
1⁄
4
∙5
121⁄3
b. (
2
)
4 1⁄3
c.
e.
g.
1
d. (24 ∙ 34 )
1
−
𝑘 3
2
3
6
(4 )
7
1
73
Additional Notes/Examples:
f. 𝑧 0
h.
2
𝑦 ⁄3
1
𝑦 ⁄3
1
4
−
Unit 1 Lesson 4 – Extending to Rational Exponents
Ready, Set, Go! - Assignment 1.4
http://goo.gl/LniWi
Name______________________________
Date_________ Hour_______
Write an equivalent form of each expression. Your answer should contain no negative variables and
each variable should only be represented once.
Ready
1.
𝑎𝑏3 ∙(𝑎𝑏)−2
2𝑥 3 𝑦 3 𝑧 2 ∙ −5𝑥 −2 𝑦 4 𝑧 2
2.
𝑎−4 𝑏3
3.
(−4𝑥𝑦 3 𝑧 3 )3
4.
5.
(𝑥 2 𝑦 0 𝑧 −2 ) ∙ (𝑥 2 𝑦 0 𝑧 4 )4
6.
3𝑥𝑦 3 𝑧 2
2−2 𝑦 −2 𝑧 4
2𝑥 −1 𝑧 3 𝑥 0 𝑦 4 𝑧 3
(
2𝑥 4 𝑧 −1
Set
Product Property of Exponents:
7.
9𝑎
5⁄
7
∙𝑎
1⁄
7
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
8.
3𝑎
1⁄
2
∙𝑎
1⁄ 2⁄
3𝑏 3
0
)
Power of a Power Property of Exponents: (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
3
2
9.
(5𝑎 2 )
11.
(243𝑥 1⁄2 𝑦 15⁄4 )
(16𝑎
10.
3
2⁄ 8 ⁄4
3𝑏 )
2⁄5
Power of a Product Property of Exponents: (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛
2
12.
(125𝑎𝑏)3
14.
(32𝑥 5 𝑦 10 𝑧)5
(16𝑥𝑦𝑧)
13.
3⁄
4
7
𝑎 𝑛
𝑎𝑛
Power of a Fraction Property of Exponents: (𝑏) = 𝑏𝑛
15.
8𝑥
5⁄
3
(𝑦)
5
17.
243𝑎2
16.
3⁄
5
(3125𝑏10 )
81𝑚 5⁄4
(
16
)
Quotient Property of Exponents:
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛
1
18.
𝑐 2⁄ 1
𝑐6
19.
𝑎3/2 𝑏3/2 𝑐 7/6
𝑎2/5 𝑐
Zero Exponent Property: 𝑎0 = 1, 𝑎 ≠ 0
3
20.
0
6 (525𝑎4 )
21.
−20
1
Negative Exponent Property: 𝑎−𝑛 = 𝑎𝑛
22.
−2𝑎
−3
23.
5𝑥 −2/3
𝑥 1/3 𝑦 −3/4
Go!
Write an equivalent form of each expression. Your answer should contain no negative variables and
each variable should only be represented once.
24. 𝑧 4 ∙ 𝑧 2
2
1
25. 𝑦 3 ∙ 𝑦 6
1
26. 𝑥 2 ∙ 𝑥 2
1
27. (32 ∙ 33 )2
30. 𝑏
0
3
1
2
28. (𝑥 4 ∙ 𝑦 6 )
31.
1
𝑚0
32.
1
−
𝑚 2
33.
3
𝑥4
36.
𝑤3
2
−
𝑤 3
34.
63
1
62
5
𝑥
3
1
29. (25 ∙ 42 )
1
36
35.
7
38
1
3
37. (𝑥 4 ∙ 𝑦 2 )
−
𝑧 2
3
−
𝑧 4
1
38. (2𝑥 5 )2
Unit 1 Lesson 5 - Perfect Roots with Variables
Notes 1.5
Fill in the following table
√1 =
√4 =
√9 =
√16 =
√25 =
√36 =
√49 =
√64 =
√81 =
√100 =
√121 =
√144 =
√169 =
√196 =
23 =
33 =
43 =
53 =
63 =
73 =
83 =
4
25 =
5
𝟑
√
=2
24 =
√16 =
√32 =
Examples
1.
3.
5.
7.
√4𝑎2
3
2.
3
√8𝑥 6
√27𝑦13
4.
√16𝑝𝑞𝑟
√25𝑒 7 𝑔12
6.
√169𝑠 2 𝑡 2 𝑢2 𝑣 2
3
√𝑥11 𝑦
8.
4
√16𝑘 8 ℎ32 𝑟 2
Unit 1 Lesson 5 - Perfect Roots with Variables
Ready, Set, Go! - Assignment 1.5
http://goo.gl/p6DBI
Name______________________________
Date_________ Hour_______
Rewrite each radical expression in an equivalent form. Radicals should contain variables that could be
rewritten outside of the root.
1.
3.
5.
7.
9.
11.
13.
√4𝑎2 𝑏 4 𝑐 7
3
2.
3
√8𝑚𝑛2
√27𝑥11 𝑦
4.
√16𝑝𝑞 22 𝑟 11
√25𝑒 2 𝑓 5 𝑔
6.
√49𝑠 2 𝑡 2 𝑢2 𝑣 2
3
√27𝑥11 𝑦
8.
4
10.
√𝑥𝑦𝑧
11
√121
5
√32𝑥10 𝑦 7 𝑧 9
12.
14.
4
√16𝑘14 ℎ12 𝑟 2
3
√64𝑢𝑤
√169𝑞 8 𝑟 5 𝑠10 𝑡
2
√144𝑎𝑏𝑐
Unit 1 Lesson 6 – Negative Roots and the Quotient Property
Notes 1.6
Negative Roots
Example 1a:
Example 1b:
vs.
√64
3
√−64
√27
Example 1c:
4
√x 4
vs.
3
vs.
√−27
Example 1d:
5
4
√−x 4
√32x 5
5
√−32x 5
vs.
The Quotient Rule for Radicals
n
n
a
√b =
√a
n
√b
Simplify each radical.
Example 2a:
√
25
9
Example 3a:
√
50
49
Example 2b:
√
15
3
Example 3b:
4
√
a5
b8
Example 2c:
3
√
b
125
Example 3c:
3
√
x5
8
Unit 1 Lesson 6 - Negative Roots and the Quotient Property
Ready, Set, Go! - Assignment 1.6
http://goo.gl/g0v7o
Name______________________________
Date_________ Hour_______
Ready
True or False.
1a.
2a.
3a.
3
√−27 = −3
4
√16 = 2
1
1
√ =
4
2
Set
Rewrite each radical in an equivalent form.
3
4.
√−1
6.
5
√−32
4
8.
− √16
10.
√
12.
𝑡
−8𝑥 6
√
√−25 = −5
2b.
√10
2
= √5
√6
= √2
3b.
√3
5.
√−1
7.
− √−8
9.
11.
4
3
1b.
𝑦3
13.
3
6
√64
𝑤
√− 36
3
−27𝑦 36
√
1000
12
√
14.
25
4
𝑥5𝑦4
√
16.
𝑧 12
15.
17.
98
√
3
9
81
√
8𝑏 3
Go!
Classify each number as Rational or Irrational and justify your reasoning.
RATIONAL
18. 5
IRRATIONAL
5
19. 7
20. 0.575
21. √5
22.
√10
2
Classify each sum or product as Rational or Irrational and justify your reasoning.
23. 5 + √7
24. 3 + 14
1
25. 4 (3)
26. √3 + √27
5 1
27. 7 (2)
28. √2(√8)
RATIONAL
IRRATIONAL
Unit 1 Lesson 7 –Radicals and Rational Exponents
Task 1.7
Name_____________________________________
Date____________ Hour________
Evaluate each of the following expressions.
93 
91 
92 
90 
What do you think the following expressions will equal?
1
1
92 
1
162 =
83 =
1
What about 152 ?
What is a “rational number”?
What is a rational exponent?
If n is any positive integer, then
There are 2 ways you can write a radical:


Radical notation
Exponential notation (rational exponents)
1
Let 𝑎𝑛 be an nth root of a, and let m be a
positive integer.
𝑚
𝑚

𝑎𝑚/𝑛 = (𝑎1/𝑛 ) = ( √𝑎)

𝑎−𝑚/𝑛 = (𝑎1/𝑛 )
−𝑚
𝑛
=
1
𝑛
𝑚
( √𝑎 )
,𝑎 ≠ 0
𝑎
1⁄
𝑛
𝑛
𝑛
= √𝑎 ,
Provided that √𝑎 is a real number.
Rewrite each expression in radical form.
1
2
1. 𝑥 3
3
2. 53
1
4. (3𝑦)6
1
7. (2𝑧)3
3. 𝑦 4
2
4
5. (𝑥𝑦)5
6. (3𝑥𝑦)5
2
8. (8𝑎𝑏)5
6
9. (−2𝑐𝑑)13
Rewrite each expression with rational exponents.
7
3
5
11. √(𝑤)3
12. √(𝑧)4
13. √(−6)2
14. 9√(2𝑥𝑦)
15. √(5𝑎𝑏)3
16. 4√(−12𝑥)
6
17. √(6𝑏)5
11
18. √(−9)7
15
19. √(15𝑎𝑏)5
Unit 1 Lesson 7 –Radicals and Rational Exponents
Notes 1.7
Notes:
Example 1
Write each radical expression using exponent notation:
a.
d.
3
35
b.
5
x
c.
5a
82
e.
( 5 x )4
f.
( 3) 2
c.
(8)1/3
Example 2
Write each exponential expression using radical notation:
2
a.
41/2
b.
93
Example 3:
Evaluate each expression:
−1⁄
2
a.
49
c.
−162
1
Example 4:
Evaluate the following expression:
4
√25𝑤 2
Additional Notes/Examples:
2
b.
325
d.
(−81)
1⁄
4
Unit 1 Lesson 7 – Meaning of Rational Exponents
Ready, Set, Go! - Assignment 1.7
http://goo.gl/murTP
Name______________________________
Date_________ Hour_______
Write each expression in radical form.
1
1
1.
𝑎4
4.
16−4
2
2.
𝑏5
5.
25−2
1
3.
𝑓3
6.
273
1
1
Write each radical in exponential form. (using rational exponents)
7.
10.
3
√𝑦
4
√16𝑧
8.
8
√𝑥
9.
11.
√23
12.
Evaluate each expression
13.
15.
16
81
1
1⁄
4
−
1
2
14.
∙ 81
3
2
25−2
1
16.
8 3
( )
27
2
√(2𝑎)3
3
√5𝑥 2 𝑦
1
17.
−3
162
1
92
18.
(−32) 4
20.
21⁄2 ∙ 21⁄3
Rewrite each problem in an equivalent form.
1⁄
3
3
21.
33 ∙ 3− 3
1
∙ 3
1⁄
4
19.
1
22.
4
√9𝑧 2
2
23.
6
√81𝑎4 𝑏8
24.
𝑟3
1
𝑟6
Unit 1 Lesson 8 –Adding and Subtracting Radicals
Notes 1.8
“Rules” to consider when adding and subtracting radicals:
Add or subtract to write an equivalent form of each expression. Write your answers with all like
terms combined. (No decimal approximations allowed, assume all variables are positive.)
Example 1:
4
4
a.
3√5 + 4√5
c.
√8 + √18
d.
√2𝑥 3 − √4𝑥 2 + 5√18𝑥 3
e.
√5 − 3√5
f.
5√7𝑥 + 4√𝑥
g.
3
√8 + √28
b.
√𝑤 − 6 √𝑤
Example 2:
a.
√2 − √8
c.
2√24 + √81
3
3
Additional Notes/Examples:
b.
√45𝑥 3 − √18𝑥 2 + √50𝑥 2 − √20𝑥 3
d.
2√45 − 3√20
Unit 1 Lesson 8 – Adding and Subtracting Radicals
Ready, Set, Go! - Assignment 1.8
http://goo.gl/xnDW3
Name______________________________
Date_________ Hour_______
Add or subtract to write an equivalent form of each expression. Write your answers with all like
terms combined. (No decimal approximations allowed, assume all variables are positive.)
1.
2.
3√6𝑎 + 7√6𝑎
√3 − 2√3
3.
5.
3
3
√4 + 3 √4
3
3
4.
3
3
√5𝑦 − 4 √5𝑦 + √𝑥 + √𝑥
6.
√2 − 5√3 − 7√2 + 9√3
4
4
√𝑎𝑏 + √𝑎 + 5√𝑎 + √𝑎𝑏
Add or subtract to write an equivalent form of each expression. Write your answers with all like
terms combined. (No decimal approximations allowed, assume all variables are positive.)
7.
√12 + √24
8.
√3 − √27
9.
3√50 − 2√32
10.
√20 − √125
11.
√12𝑥 5 − √18𝑥 − √300𝑥 5 + √98𝑥
13.
5√24 + 2√375
15.
17.
3
3
5
14.
5
√64 + 7√2
3
16.
3
√2000𝑤 2 𝑧 5 − √16𝑥 2 𝑧 5
True or False. If false, give the correct value.
19.
√3 + √3 = √6
21.
12.
2√5 + 3√5 = 5√10
3
3
3
3
2√24 + √81 + √27 − √−8
4
4
√48 − 2√243
3
3
√54𝑡 4 𝑦 3 − √16𝑡 4 𝑦 3
18.
2√45 − 3√20
20.
√8 + √2 = 3√2
22.
√12 = 2√6
Unit 1 Lesson 9 –Multiplying Radicals
Notes 1.9
Rules” when multiplying radicals:
Multiply each radical expression. Assume all variables are positive.
a.
5√6 ∙ 4√3
b.
c.
3√2 (4√2 − √3)
d.
e.
(2√3 + √5)(3√3 − 2√5)
4
𝑥3
√
3
2
4
3
3
4
√2 ∙ √2
b.
3
√𝑎 ( √𝑎 − √𝑎2 )
Example 2:
a.
𝑥2
∙ √8
3
√2 ∙ √3
Example 3:
a.
(2 + 3√5)(2 − 3√5)
Additional Notes/Examples:
b.
√ 3√𝑥
Unit 1 Lesson 9 – Multiplying Radicals
Ready, Set, Go! - Assignment 1.9
http://goo.gl/clHt5
Name______________________________
Date_________ Hour_______
Multiply each radical expression. Assume all variables are positive. Give exact answers.
Ready
1.
√5 ∙ √7
3.
5.
(3√2)(−4√10)
4
4
2.
2√5 ∙ 3√10
4.
2√7𝑎 ∙ 3√2𝑎
√9 ∙ √27
6.
2 6( 24  7)
8.
3
3
√5 ∙ √100
Set
7.
9.
( 75  8) 12)
3 5(  20  2)
10. (3 12  4) 27)
11. 4 2( 12  3 2  4 8)
3
3
3
Go!
13. (2  3)(1  3)
15. (8  12)(3  12)
17. ( 100  6)(4  12)
3
12. 5√3( √9 − 4 √18 + 2 √9)
14. ( 5  8)(1  3 5)
16. (6  5)(4  2 5)
18. (4  2 27)(1  75)
Additional Multiplying Radicals Practice
Worksheet/Task
Name_______________________________
Date______ Hour______
Multiply and write each in an equivalent form.
3
3
1.
√5 ∙ √3
2.
3.
−4√28 ∙ √7
4.
3√12 ∙ √6
6.
3√3(4 − 3√5)
8.
√15(2√10 − 4√6)
5. √6(√3 + √12)
7.
9.
4√15(−3√6 + 5)
3
3
3
√4( √2 − √5)
√3 ∙ √−20
3
3
10. (√2 − 5)(7√2 − 5)
4
11. (−2√3 + 2)(√3 − 5)
3
8
6
3
14. √ √𝑧
13. √10 ∙ √10
15.
17.
7
12. √2 ∙ √2
10
3
√√2
5
16. √6 ∙ √6
4 7
√ √𝑎
18.
6
√5 ∙ √5
19.
√ 4√3
20.
Find the area of a rectangle whose length is √20 inches, and whose width is √20 inches.
5
Unit 1 Lesson 10 – Quotients and Rationalizing Denominators
Notes 1.10
Square roots such as √2, √3, 𝑎𝑛𝑑 √5 are irrational numbers. If roots of this type appear in the denominator of a
fraction, it is customary to rewrite the fraction with a rational number in the denominator, or rationalize it.
Rewrite each expression with a rational denominator.
Example 1:
a.
√3
√5
b.
3
3
√2
Simplified Radical Form for Radicals of Index n
A radical expression of index n is in simplified radical form if it has
1. NO perfect root radicands
2. NO fractions inside the radical
3. NO radicals in the denominator.
4. Exponents in the radicand and the index of the radical have
greatest common factor 1
Simplify.
Example 2:
a.
c.
√10
√6
𝑎
√𝑏
b.
d.
5
3
√
9
√
𝑥3
𝑦5
Example 3:
a.
3√2
2√3
b.
3
10𝑥 2
√
5𝑥
Example 4:
a.
4−√12
4
b.
−6+√20
−2
Unit 1 Lesson 10 – Quotients and Rationalizing Denominators
Ready, Set, Go! - Assignment 1.10
http://goo.gl/Z8Aua
Name______________________________
Date_________ Hour_______
True or False. If false, give the correct value.
1.
4.
√6
√2
= √3
4−√10
2
= 2 − √10
2.
5.
2
√2
= √2
8√7
2√7
1
3.
= 4√7
√3
√12
3
6.
=
√3
3
= √4
Rewrite each expression with a rational denominator.
7.
2
8.
√5
√3
9.
√7
7
3
√3
Write each radical expression in simplified radical form. (Assume all variables are positive)
10.
12.
14.
7
√
18
3
7
√
4
2
5𝑥
√2𝑦
11.
13.
15.
2
3
√
3
√𝑥 5
√𝑦 3
3
3
√
4𝑎2
Divide and write each in an equivalent form.
16.
(3√3) ÷ (5√6)
17.
18.
√32𝑥 3⁄
√48𝑥 2
19.
4
3
20.
22.
24.
√20
3
√2
10+√50
5
−6+√72
−6
(5√12) ÷ (4√6)
21.
23.
√48⁄4
√3
6+√45
3
−2+√12
−2
Unit 1 Lesson 11 – Rational Expressions Concept Check/Review
Task 1.11
Name______________________________
Date_________ Hour_______
State whether the given equations are TRUE or FALSE. Be ready to justify your responses.
a. 16
1⁄
4
=4
1⁄
2
1
c.
e.
g.
i.
52 ∙ 52 = 254
f.
3
3𝑥
𝑥+1
=3
2+√6
√16 = √4
h.
2 − √−6 = 2 − 6𝑖
j.
√2 = √2
𝑙.
√12 = 2√6
1
k.
3
d. √9 = 3
42 = √2
6
3
(√2) = 2√2
b.
6− 2 =
√6
6
2
= 1 + √6
2
√2 ∙ √2 = 2
m.
o.
3
√−27 = −3
n.
p.
3
√2 ∙ √2 = 2
√−25 = −5
r.
q.
9
√17 ∙ √17 = 289
4
3
=2
t.
s.
√18 = 9√2
√283 = 17
v.
u.
√9 + √16 = √25
√4 = √2
w.
3
√15
3
= √3
y. √5 + 3√5 = 4√10
x. (√3 − 1)(√3 + 1) = 2
z.
3
√525 = 15
Unit 1 Lesson 11 – Rational Expressions Concept Check/Review
Ready, Set, Go! - Assignment 1.11
http://goo.gl/4qfk1
Name______________________________
Date_________ Hour_______
Directions: Using ALL FALSE problems from the task, find the correct answer to each. Be sure to
show your work and label each problem. Then, write 10 true/false problems of your own.
Unit 1 Review – Expanding Our Number System
Review Assignment
Name______________________________
Date ________ Hour______
1.
Reorder each group of numbers from smallest to largest.
3
2
1
7 10 3
50, −3, − 75.2, (− ) , −1, − , 𝜋, , ,
4
3
3
8 5 5
2.
3.
4.
Give, if possible, an example of each type of number. If not possible, explain why.
a.
a negative real number that is not rational.
b.
a negative real number that is not irrational.
c.
a negative integer that is not real.
d.
a negative real number that is not an integer.
Interpret 10 and -4 in these settings.
a.
a financial situation of some sort
b.
a sport
c.
a temperature change
d.
a temperature
Give eight fractions between each pair of numbers.
a.
7
4
9
𝑎𝑛𝑑 5
b.
5.
Is the set of even integers closed under addition? Why or why not?
6.
Is the set of multiples of 3 closed under addition? Why or why not?
2
31
(− 3) 𝑎𝑛𝑑 (− 50)
7.
Is the set of integers closed under subtraction? Why or why not?
Simplify each radical. Assume variables represent positive numbers. Write answers in exact and reduced
form. (no decimal approximations)
8.
162
9.
156
10.
350
11. 7 40
12. 12x 3 y 8 z 2
13. 6 25x 4 y 0 z 6
14. 8 5  10
15.  8 3  2 300  4 27
16.
3 8  2 125  4 20  98
18. √10 ∙ √6
19. 3√12 ∙ 2√30
17. 6 5a7 
45a3
20. 9√3𝑐 ∙ 3√15𝑐
21. √𝑐 5 𝑑 2 𝑒𝑓 4 ∙ √𝑐 2 𝑒 6 𝑓 4
22.

 
25n  2n  n 18
Simplify the following. Write answers with only positive exponents.
23.
51 + 50 + 5 –1
24.
25.
85
87
26.
27.
(3)
28.
(2−3 )2
29.
2 –2 + 4 –2
30.
(2 + 4) –2
31.
(2𝑥 −1 )3 ∙ 𝑥
32.
(2𝑥 − 4)0 + 812
33.
5 −2
−
8
2
3
(−2𝑥)−2
1
4−2
1
34.
1
2
(x ) 2

Unit 1 Lesson 12 – HONORS – Rationalizing Expressions
Notes 1.12
1a. What is the result when we multiply the square root of a real
number by itself?
Example: √5 ∙ √5
Product property of
radicals
If n is even and a and b are both
non-negative, then
n
ab  n a  n b
If n is odd, then
n
ab  n a  n b
b. What is another term for “multiply something by itself?”
c. Notice that the last example of binomial form multiplication cleaned up very nicely. Can you
explain why?
d. Define CONJUGATES
We can use conjugates to “rationalize” denominators—that means we will make the denominator a
rational number, no radicals or rational exponents..
2.
3
3
Multiply: √5 ∙ √5
Compare your answer to the previous problem. Explain why the radical did not cancel in this problem.
3.
What do you think you would need to do to “un-do” a cube root? How about a fourth root? Refer to #1
for a hint
Quotient property of
radicals
For any real numbers a and b not
equal to zero, and any integer n >
1,
n
a na

b nb
, if all roots are
Examples
Single term
27
3
Binomial form
3
2 1
defined.
To simplify:

n must be as small as
possible

The radicand contains not
factors that are nth powers
of an integer or
polynomial.

The radicand contains no
fractions

No radicals or negative
powers appear in
denominators.
Additional Examples:
x4
y6
2 5
1 3
Unit 1 Lesson 12 HONORS – Quotients and Rationalizing
Denominators Ready, Set, Go! - Assignment 1.12
http://goo.gl/UOsjH
Name______________________________
Date_________ Hour_______
Ready
Write an equivalent form of each.
48x 3 y 8
125
2.
3
3. 3 3  2 20  2 3
4.
24 x 3 yz 5
16 x 2 y 4 z 2
8.
5  5 3  5 
1.
Set
Multiply.
7.
9.

7 2  11
3 


17 3  17

10.
2

34 5 2 3 4 5

Divide. Make sure all denominators are rationalized.
11.
1
5
12.
1
5
3
13.
5
5
4
14.
2x
3x
17.
3 2
2 4
15.
4b
3
16.
2b 2
55 2
35 2
18.
Go!
Write each in an equivalent form.
3
2
19. √125 ∙ √125
20.
1
1
x4
21.
x
22.
1
1
1
1
a2  b2
a2  b2
6
2 3