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Algebra 1: Chapter 10 Notes
1
Algebra Homework: Chapter 10
(Homework is listed by date assigned; homework is due the following class period)
Leave all answers in reduced, radical form. No decimals please!!!
HW#
15
16
Date
F
5/6
M
5/9
17
T
5/10
18
W
5/11
19
Th
5/12
20
F
5/13
21
M
5/16
In-Class
Section 10.1: Simplifying and Multiplying Radicals
Section 10.1 and 10.2: Dividing Radicals and the
Pythagorean Theorem
Section 10.3: Adding, Subtracting, and Distributing
Radical Expressions
Review of Sections 10.1-10.3
Section 10.4: Solving Radical Equations
Chapter 10 Review
Chapter 10 Test
Homework
HW#15
Pg. 489: 2-24 even
Pg. 580: 41-48 all
Get Progress Report Signed by
Wednesday
HW#16
Pg. 489: 15-23 odd, 28-50 even
Pg. 496: 1-3, 7-9, 16-18
Get Progress Report Signed by
Wednesday
HW#17
Pg. 490: 33-51 odd
Pg. 496: 10, 11, 20
Pg. 503: 1-3, 10-31x3
HW#18
Pg. 521: 10-48 even
Pg. 580: 58-64 all
HW#19
Pg. 506: 1-10 all
Pg. 524: 1-8 all
Pg. 510: 1-3, 9-12, 38, 45, 46
(Show ALL checks on pg. 510)
Chapter 10 Test and Notes
Check on Tuesday
HW#20
Chapter 10 Study Guide
Correct Study Guide Online
Chapter 10 Test and Notes
Check on Tuesday
HW#21
Pg. 580: 49-52, 55, 65-72
Print: Chapter 9 Notes
1
Algebra 1: Chapter 10 Notes
Notes #15: Real and Radical Expressions (Sections 10.1)
A. Square Roots
Complete:
2
02 = ____
12 = ____
22 = ____
32 = ____
42 = ____
52 = ____
62 = ____
72 = ____
82 = ____
92 = ____
102 = ____
112 = ____
122 = ____
132 = ____
142 = ____
152 = ____
202 = ____
302 = ____
402 = ____
502 = ____
The numbers you just wrote down are called ___________ __________ because you _______________
another number to find them.
Complete:
What number squared makes 16? _____ or _____
What number squared makes 81? _____ or ______
What number squared makes 0? ______
What number squared makes -4? ______
You just found the ___________ _________ (
) of each of these numbers. Positive numbers have
______ real square roots and negative numbers have _______ real square roots.
How do you know which root to list, the positive, the negative, or both?



25 means the _____________ (or positive) square root of 25.
25 = ____
- 25 means the ______________ square root of 25.
- 25 = ____
 25 means ___________ square roots of 25.
 25 = ____ or ____
Find the square roots of each number:
1.) 81
2.) 64
3.) 100
Simplify:
4.)

9
5.)
8.) Evaluate 2 x  7 for x = 2.
Is this a real number?
49
6.)
 121
7.)
196
9.) Evaluate 4  8y for y = -1. Is
this a real number?
2
Algebra 1: Chapter 10 Notes
Simplify: (if it is a polynomial, factor first!!)
10.)
14.)
m2
16b2
11.)
x2  4 x  4
3
15.)
13.)
4d 2 12d  9
(2b  5) 2
16.)
18
B. Simplifying Square Roots
For problems like #16, you use one of two methods:
(1) Find perfect square factors of 18
to help you break it down
(2) Write a complete factor tree
for 18 and simplify by taking out
“buddies”
18
18
Simplify each expression using BOTH methods. Then decide which you prefer:
Assume that all variables are nonnegative. (If it is a polynomial - ____________ first!!)
1.)
24
2.)
4.)
48a 2
5.)
300
125b3
3.)
6.)
50m
42g 5
3
Algebra 1: Chapter 10 Notes
x7
7.)
10.)
4
8.)
8( w  3) 4
(m  5)5
11.)
9.)
200(k  2)5
12.)
5x2  10 x  5
150a3b7
B. Multiplying Square Roots
We will be multiplying expressions like: (2
Steps: - simplify each radical completely
20)(3 5)
- (outside • outside) inside inside or
- simplify your answer, if possible
(a b )(c d )  ________
*Remember, if you are multiplying polynomials, you must ________**
Multiply and simplify, if possible:
1.)
5
6
2.)
4.)
4
x
5.) 2 3
7
7
4 6
3.)
64
6.) 2 2
7
5 10
4
Algebra 1: Chapter 10 Notes
7.) 3 6a
3
10.)
7a
13.)
15c2d 3
4 8a
14b
20c3d 4
5
3
8.) 5 12 x y
11.)
14.)
3 2 
-2 10 xy 5
2
2m2 n
9.) 3 6 x
2 3
3a3b2c
12.) 12a b c
10m3n

15.) 5 7
2 10 x

2
5
Algebra 1: Chapter 10 Notes
6
Notes #16: Dividing Radical Expressions and the Pythagorean Theorem (Sections 10.1 and 10.2)
Section 10.1: Dividing Rational Expressions
Taking the square root of a fraction is the same as taking the square root of the ___________________
and ______________________ separately
OR
You can ___________ first and then split up the fraction
Examples:
9

16
12

27



Simplify:
1.)
4.)

81
100
6
24
2.)
121
196
3.)
5.)
75
300
20a 3
6.) 
80a
8m3
7.)
2m5
3x3
25
144
60 y 7
24 x 4
8.)

9.)
3 y3
6
Algebra 1: Chapter 10 Notes
7
However, sometimes our fractions don’t simplify as well…we end up with a radical expression in
the denominator. This is NOT allowed!!
Ex:
24
30
Ex:
6x2
12
To fix this problem:
- simplify the fraction as much as you can
- multiply this simplified fraction (both ________ AND __________) by the exact
term that is still in a square root sign on the denominator
- simplify and reduce
Simplify:
10.)
12.)
14.)
16.)
5
6
11.)
1
5
5
x
12
2
13.)
15.)
17.)
3
8
2c
6c3
3
2x
d3
27
7
Algebra 1: Chapter 10 Notes
18.)
20.)
14
3x
8
19.)
18
18
x4
108 xy 5

2x
21.)
y3
Section 10.2: The Pythagorean Theorem
Solve for x:
1.) 32 + 42 = x2
2.) 132 = 122 + x2
3.) x2 + 42 = 82
Right Triangles: Triangles with one _____________ ______________.
hypotenuse = ____________________
legs = __________________________
Pythagorean Theorem:
( _________)2 + (_________)2 = (___________________)2
OR
a2 + b2 = c2
8
Algebra 1: Chapter 10 Notes
9
Use Pythagorean Theorem to solve for the third side of each right triangle. Leave your answer in
simplified radical form:
4.)
5.)
b
8
c
3
16
4
6.)
7.)
x
x
x
2 3
6
4
8.) a = 5, b = 12, c = ?
9.) a = 1, c =
10.) A 15 foot ladder is leaning against a building.
The bottom of the ladder is 9ft from the building.
How high is the top of the ladder?
11.) How long must a wire be to reach from the top
of a 12-m telephone pole to a point on the ground
5m from the foot of the pole?
3,b=?
9
Algebra 1: Chapter 10 Notes
10
Determine whether the given lengths can be sides of a right triangle.
- use the longest length as c. Use the shorter two lengths as a and b
- plug into the Pythagorean theorem: a2 + b2 = c2
- if both sides of the equation are equal, then the triangle is ________
If both sides of the equation are not equal, then the triangle is ______ _____________
12.) 2ft, 3ft, 4ft
13.) 6in, 7in, 8in
14.) 5cm, 5cm, 5 2 cm
Notes #17: Other Operations on Radical Expressions (Section 10.3)
Adding and subtracting square roots is the same process as adding and subtracting ________________
________________: look for ________________!!
Review: Add/Subtract
1.) 3x – 2y – 8x + 7y
2.) -2mn – (-3x2) + mn – 7x2
Steps for adding/subtracting radical expressions:
 _______________ all radical expressions (break each term down as far as possible)
 Look for _________________ (underline, circle, box, etc)
 Combine like terms. Add the _______________, but leave the roots __________
Add/Subtract:
1.)
4.)
2 5 3 5
8  18
2.)
6 7  (3 7)
5.)
3.)
12 3  5 2
3 12  6 27
10
Algebra 1: Chapter 10 Notes
11
2 40  (3 90)
6.)
7 12 x  3 48x
8.)
2
10.)
13.)

2 3 4 6
7.)
9.)
2

 2  3  4  5 3 
11.) 2 3
14.)

x  4x
3 12  4 20  3 45  (2 300)
3 5 8

 6  5  6  5 

5 4 5
12.)


13.) 3 2  1

2
What was special about #14?
Steps for simplifying a fraction with a binomial in the denominator:
 Multiply the _______ and ____________ of the fraction by the conjugate
3
2
Ex:
Ex:
2 5
3 5


Distribute in the numerator, use ________ in the denominator
Reduce only if ________ terms can be simplified by the same factor
11
Algebra 1: Chapter 10 Notes
12
3
2 5
Ex:
Ex:
2
3 5
2.)
6
10  2
4.)
15
8 5
Simplify
1.)
8
4 6
13
7 5
3.)
Notes #18: Review of Sections 10.1 – 10.3
Simplify:
1.)
48a3b7
4.) 6
12( x  1) 2
6.)
 5 2  2 12 
2.) 5 x 2 y 24 x 3 y 4
3.) 4m2 n 63mn5
4 y 2  40 y  100
5.)

7.) 12 j 5 k 8 j 2 k
 3 jk
2
20 jk 4

12
Algebra 1: Chapter 10 Notes
48m5
3m
8.)
9.)

11.) 3 2 6  2

14.) 1  7
16.)
13


8
2

10.)
12.) 5 6

7
5 3
19.) 4 5  125  40

2

13.) 1  3 2  4 3
15.)
5
12  2
17.)
37 3
2
18 x 2
5
18.)

12  5 27


20.) 3 18  6 72
13
Algebra 1: Chapter 10 Notes
Notes #19
Review Concepts First: Solve for x
1.) x 2  42  52
2.) x 2  42  82
4.)
x  12
5.)
14
3.) x 2  12
x  3  12
6.)
x  3  12
Section 10.4: Solving Equations with radical expressions
Steps:
 Get the
all alone
2
 Square ( ) both sides (If you square a binomial, be sure to use _______!!)
 Solve for x
 Plug it back to check for extraneous solutions (you can never take the square root of a
______________ number)
Solve for x:
1.)
3.)
x 8
2x  4  7
2.)
x 1  3
4.)
x 1  2x  5
14
Algebra 1: Chapter 10 Notes
15
5.)
3x  5  3
6.)
7.)
x 2 5  3
8.)
3m  6  2m  3
10.)
x  2  x2  8
9.) 4  10  5 x  11
x 1  x2  5
15
Algebra 1: Chapter 10 Notes
Classwork Day #20: Chapter 10 Review
Simplify each radical expression.
1.
56x8 y 7
5.
5 3 
12.
2. 8a 2 108a9
2
9. 2 48  10 12
7 10  3 6
2
16
6.
7 3
6x

10. 6 5 7  3 10
13.
3.
81m8
100n16
4. 5ab 6a2b3
7.
12c
48b 4
8.

11.
6


2a 10ab
7 5  3 5
3 5 5 7 3  5


12
6 3
16
Algebra 1: Chapter 10 Notes
Solve for the variable:
14. a2 + 62 = 122
17
15.
5
3
b
16.
x2  108  2x
17. 3  6  x  2
17
Algebra 1: Chapter 10 Notes
Algebra 1: Chapter 10 Study Guide
18
Name: __________________
Simplify each radical expression.
1.
48x 6 y 5
5.
49 y 8
64 x 6
6.
9.
48b
81
10.
2.
112m3n7
6 12 3 6
5 2
6m
3. 6a 72a5
7. 3 g 2 g 2 h
11.
56b
24c 4
4. 5d
2h 10 gh
8.
12.
121
64
7 5 

2
5 11  6 11

18
Algebra 1: Chapter 10 Notes
19

14. 3 32  7 8
13. 2 28  6 63
Solve for the variable:
16. a2 + 42 = 52
17. 52 + b2 = 102

15.
d3
27
18.
15
b
9
Simplify:

19. 5 2 7  2 6
22.
6
32

20.
4

32 2 5 3 2
23.

21.
5 15  3 2
3
6
3 7
19
Algebra 1: Chapter 10 Notes
Solve each radical equation and check your answer(s)
24.
x2  75  2 x
25.
x 3  2  7
20
26. 5  7  x  5
20