Chapter 9 Section 3 Adding, Subtracting and Multiplying Square Roots
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Chapter 9 Section 3
Adding, Subtracting and Multiplying Square Roots
Learning Objective
1. Add and subtract square roots
2. Multiply square roots
Key Vocabulary
• like square roots
• unlike square roots
• product rule for square roots
Add and Subtract Square Roots• Like square roots are square roots having the same radicands. They
are added in the same way as other like terms.– Adding the coefficient is an application of the distributive
property– Multiply the sum by the like square root
• Adding like terms5x + 4x = (5 + 4)x = 9x
Example: Example: Example:7 = 3 + 4 7x = 3x + 4x7 = 7 7x = (3 + 4)x
7x = 7x
7 3 3 3 4 3
7 3 (3 4) 3
7 3 7 3
Example:
Example:
Add and Subtract Square Roots
2 5 3 10 5 4
2 5 10 5 3 4
(2 10) 5 3 4
8 5 7
x x
x x
x
x
4 6 10 2 already simplified
Example:
Example:
Add and Subtract Square Roots
17 5 4 5
(17 4) 5
21 5
12 11 7 11
12 7 11
5 11
Example:
Example:
Add and Subtract Square Roots
3 9
3 9
(3 9)
12
a a a
a a a
a a
a a
5 2
5 2
1 5 2
6 2
m m m
m m m
m m
m m
Example:
Example:
Example:
Add and Subtract Square Roots
7 2 4 2 5 2
(7 4 5) 2
8 2
5 7
(5 7)
12
m m
m
m
3 5 6
( 3 5 6)
2
x x x
x
x
Unlike square roots have different radicands.
If possible change the unlike term to a like term
Sometimes we can combine by simplifying first.
Example: Example:
Add and Subtract Unlike Square Roots
4 3 48
4 3 16 3
4 3 4 30
2 50
2 25 2
2 5 2
(1 5) 2
6 2
Example: Example:
Add and Subtract Unlike Square Roots
2 8 5 18 7
2 4 2 5 9 2 7
(2)(2) 2 (5)(3) 2 7
4 2 15 2 7
(4 15) 2 7
19 2 7
20 45
4 5 9 5
2 5 3 5
(2 3) 5
5
Example: Example:
Add and Subtract Unlike Square Roots
4 3 6 2 15 3
(4 15) 3 6 2
6 2 11 3
4 3 3 6
(4 3) ( 3 6)
7 9
p q p q
p q
p q
5 2 8
(1 8) 2 5
9 2 5
a b a
a b
a b
Example: Example:
Example: Example:
Add and Subtract Unlike Square Roots
54 24 6
9 6 4 6 6
3 6 2 6 6
(3 2 1) 6
4 6
12 27
4 3 9 3
2 3 3 3
(2 3) 3
5 3
Example: Example:
Add and Subtract Unlike Square Roots
15 20 7 45
15 4 5 7 9 5
(15)(2) 5 (7)(3) 5
30 5 21 5
(30 21) 5
51 5
10 50 7 18
10 25 2 7 9 2
(10)(5) 2 (7)(3) 2
50 2 21 2
(50 21) 2
29 2
When multiplying square roots we need to use the distributive property.
x (x + 2) = x2 + 2x
Example:
Multiplying Square Roots
2
3 5
3 3 5
3 15
3 15
x x y
x x x y
x xy
x xy
Example: Example:
Multiplying Square Roots
5 2 5
5 2 5 5
10 25
10 5
2
3 2 11
3 2 11
3 2 11
3 2 11
a a
a a a
a a
a a
Example:
Multiplying Square Roots
2
2 17
2 17
2 17
m m m
m m m m
m m m
When multiplying binomials with square roots. Each term in the first binomial is multiplied by each term in the second binomial. (FOIL)
Example:
Multiplying Square Roots
4 2 3 5 2
(4)(3) (4)( 5) 2 3 2 2 5 2
12 20 2 3 2 5 4
12 (20 3) 2 (5)(2)
12 17 2 10
2 17 2
Example:
Multiplying Square Roots
3 2 4 5
(3)(4) (3) 5 2 (4) 2 5
12 3 5 4 2 10
3 5 4 2 10 12
2
8 3
(8)(3) (8) (3)
24 8 3
24 8 3
24 ( 8 3)
24 5
p p
p p p p
p p p
p p p
p p
p p
Example:
Example:
Multiplying Square Roots
2 2
2
2
4 3 5
4 3 4 ( 5 ) ( ) 3 5
12 20 3 5
12 (20 3 ) 5
12 17 5
r s r s
r r r s s r s s
r s r s r s
r s s r s
r s r s
Example:
Multiplying Square Roots
2 2
2
2
12 3 5 2 3
12 5 12 2 3 3 5 3 2 3
60 24 3 5 3 2 9
60 24 5 3 (2)(3)
60 29 3 6
x y x y
x x x y y x y y
x x y x y y
x x x y y
x x y y
Difference in Two Squares:
Example:
OR
Multiplying Square Roots
5 3 5 3
5 5 5 3 3 5 3 3
25 15 15 9
5 15 15 32
2 25 3
25 95 32
2 2 ( )( )a b a b a b
5 and 3a b
Example:
OR
Multiplying Square Roots
2 2
2
2
m n m n
m m m n n m n n
m n m n m n
m n m n m n
m n
2 2
2 2
2
a m b n
m n
m n
m n
Example: Example:
Multiplying Square Roots
22
7 9 7 9
7 9
49 (9)
7 8174
a b
22
4 4
4
(4)
16
y y
a y b
y
y
Remember• Like square roots are square roots having the same radicands
they are added in the same way as other like terms
• It may be helpful for you to review combining like terms from Chapter 2 Section 1.– Determine which terms are alike– Add or Subtract the coefficient of the like terms– Multiply the coefficient by the variable.
• Make sure you add or subtract the coefficient not multiplying them
Remember• When multiplying square roots we need to use the
distributive property
• When multiplying binomials with square roots. Each term in the first binomial is multiplied by each term in the second binomial. (FOIL)
HOMEWORK 9.3
Page 551 - 552:
# 5, 9, 13, 19, 27, 37, 53, 61, 71, 79
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