algebra1 multiplying and dividing radical expressions
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CONFIDENTIAL 1
Algebra1Algebra1
Multiplying and Multiplying and DividingDividing
Radical ExpressionsRadical Expressions
CONFIDENTIAL 2
Warm UpWarm Up
1) 6√(10) 2) 3√2 23) 7x 8y2
4) 5a2√2 3
1) √(360)
2) √(72) √(16)
3) √(49x2) √(64y4)
4) √(50a7) √(9a3)
Simplify. All variables represent nonnegative numbers.
CONFIDENTIAL 3
Multiplying Square RootsMultiplying Square Roots
Multiply. Write each product in simplest form.
A) √3√6
= √{(3)6}
= √(18)
= √{(9)2}
= √9√2
= 3√2
Multiply the factors in the radicand.
Product Property of Square Roots
Factor 18 using a perfect-square factor.
Product Property of Square Roots
Simplify.
CONFIDENTIAL 4
B) (5√3)2
= (5√3)(5√3)
= 5(5).√3√3
= 25√{(3)3}
= 25√9
= 25(3)
= 75
Commutative Property of Multiplication
Expand the expression.
Product Property of Square Roots
Simplify the radicand.
Simplify the square root.
Multiply.
CONFIDENTIAL 5
C) 2√(8x)√(4x)
= 2√{(8x)(4x)}
= 2√(32x2)
= 2√{(16)(2)(x2)}
= 2√(16)√2√(x2)
= 2(4).√2.(x)
= 8x√2
Product Property of Square Roots
Multiply the factors in the radicand.
Factor 32 using a perfect-square factor.
Product Property of Square Roots.
CONFIDENTIAL 6
Now you try!
1a) 5√21b) 631c) 2m√7
Multiply. Write each product in simplest form.
1a) √5√(10)
1b) (3√7)2
1c) √(2m) + √(14m)
CONFIDENTIAL 7
Using the Distributive PropertyUsing the Distributive Property
Multiply. Write each product in simplest form.
A) √2{(5 + √(12)}
= √2.(5) + √2.(12)
= 5√2 + √{2.(12)}
= 5√2 + √(24)
= 5√2 + √{(4)(6)}
= 5√2 + √4√6
= 5√2 + 2√6
Product Property of Square Roots.
Distribute √2.
Multiply the factors in the second radicand.
Factor 24 using a perfect-square factor.
Simplify.
Product Property of Square Roots
CONFIDENTIAL 8
B) √3(√3 - √5)
= √3.√3 - √3.√5
= √{3.(3)} - √{3.(5)}
= √9 - √(15)
= 3 - √(15)
Product Property of Square Roots.
Distribute √3.
Simplify the radicands.
Simplify.
CONFIDENTIAL 9
Now you try!
2a) 4√3 - 3√62b) 5√2 + 4√(15)2c) 7√k - 5√(7)k2d) 150 - 20√5
Multiply. Write each product in simplest form.
2a) √6(√8 – 3)
2b) √5{√(10) + 4√3}
2c) √(7k)√7 – 5)
2d) 5√5(-4 + 6√5)
CONFIDENTIAL 10
In the previous chapter, you learned to multiply binomials by using the FOIL method. The same
method can be used to multiply square-root expressions that contain two terms.
CONFIDENTIAL 11
Multiplying Sums and Differences of Multiplying Sums and Differences of RadicalsRadicals
Multiply. Write each product in simplest form.
A) (4 + √5)(3 - √5)
= 12 - 4√5 + 3√5 – 5
= 7 - √5
B) (√7 - 5)2
= (√7 - 5) (√7 - 5)
= 7 - 5√7 - 5√7 + 25
= 32 - 10√7
Simplify by combining like terms.
Use the FOIL method.
Expand the expression.
Simplify by combining like terms.
Use the FOIL method.
CONFIDENTIAL 12
Multiply. Write each product in simplest form.
Now you try!
3a) (3 + √3)(8 - √3)
3b) (9 + √2)2
3c) (3 - √2)2
3d) (4 - √3)(√3 + 5)
3a) 21 + 5√33b) 83 + 18√23c) 11 - 6√23d) 17 - √3
CONFIDENTIAL 13
A quotient with a square root in the denominator is not simplified.
To simplify these expressions, multiply by a form of 1 to get a perfect-square radicand in the
denominator. This is called rationalizing the denominator.
CONFIDENTIAL 14
Rationalizing the DenominatorRationalizing the Denominator
Simplify each quotient.
A) √7 √2
= √7 . (√2) √2 (√2)
= √(14) √4
= √(14) 2
Product Property of Square Roots
Multiply by a form of 1 to get a perfect-square radicand in the denominator.
Simplify the denominator.
CONFIDENTIAL 15
B) √7 √(8n)
= √7 √{4(2n)}
= √7 2√(2n)
= √7 . √(2n) 2√(2n) √(2n)
= √(14n) 2√(2n2)
= √(14n) 2 (2n)
= √(14n) 4n
Simplify the denominator.
Write 8n using a perfect-square factor.
Multiply by a form of 1 to get a perfect-square radicand in the denominator.
Simplify the square root in the denominator.
Product Property of Square Roots
Simplify the denominator.
CONFIDENTIAL 16
Simplify each quotient.
Now you try!
4a) √(13) √5
4b) √(7a) √(12)
4c) 2√(80) √7
4a) √(65) 5
4b) √(21a) 6
4c) 8√(35) 7
CONFIDENTIAL 17
Assessment
1) √2√3
2) √3√8
3) (5√2)2
4) 3√(3a)√(10)
5) 2√(15p)√(3p)
1 )√62 )2√63 )125
4 )3(√30a)5 )6p√5
Multiply. Write each product in simplest form.
CONFIDENTIAL 18
6 )2√6( + √42)7 )5√3 - 3
8( )√35( - )√21)9 )2√5 + 16
10 )5(√3y + )4√(5y)
6) √6(2 + √7)
7) √3(5 - √3)
8) √7{√5 - √3)
9) √2{√(10) - 8√2}
10)√(5y){√(15) + 4}
Multiply. Write each product in simplest form.
CONFIDENTIAL 19
11 )12 - 7√212 )6 - √6
13- )5 - √2√314 )28 + 10√315 )54 + 36√2
11)(2 + √2) (5 + √2)
12)(4 + √6) (3 - √6)
13)(√3 - 4) (√3 + 2)
14)(5 + √3)2
15)(√6 - 5√3)2
Multiply. Write each product in simplest form.
CONFIDENTIAL 20
Simplify each quotient.
16) √(20) √8
17) √(11) 6√3
18) √(28) √(3s)
19) √3 √6
20) √3 √x
16 )(√10) √2
17 )(√33) 18
18 )2(√21s) 3s 19 )√6
2 20 )(√3x)
x
CONFIDENTIAL 21
Multiplying Square RootsMultiplying Square Roots
Multiply. Write each product in simplest form.
A) √3√6
= √{(3)6}
= √(18)
= √{(9)2}
= √9√2
= 3√2
Multiply the factors in the radicand.
Product Property of Square Roots
Factor 18 using a perfect-square factor.
Product Property of Square Roots
Simplify.
Let’s review
CONFIDENTIAL 22
B) (5√3)2
= (5√3)(5√3)
= 5(5).√3√3
= 25√{(3)3}
= 25√9
= 25(3)
= 75
Commutative Property of Multiplication
Expand the expression.
Product Property of Square Roots
Simplify the radicand.
Simplify the square root.
Multiply.
CONFIDENTIAL 23
Using the Distributive PropertyUsing the Distributive Property
Multiply. Write each product in simplest form.
A) √2{(5 + √(12)}
= √2.(5) + √2.(12)
= 5√2 + √{2.(12)}
= 5√2 + √(24)
= 5√2 + √{(4)(6)}
= 5√2 + √4√6
= 5√2 + 2√6
Product Property of Square Roots.
Distribute √2.
Multiply the factors in the second radicand.
Factor 24 using a perfect-square factor.
Simplify.
Product Property of Square Roots
CONFIDENTIAL 24
In the previous chapter, you learned to multiply binomials by using the FOIL method. The same
method can be used to multiply square-root expressions that contain two terms.
CONFIDENTIAL 25
Multiplying Sums and Differences of Multiplying Sums and Differences of RadicalsRadicals
Multiply. Write each product in simplest form.
A) (4 + √5)(3 - √5)
= 12 - 4√5 + 3√5 – 5
= 7 - √5
B) (√7 - 5)2
= (√7 - 5) (√7 - 5)
= 7 - 5√7 - 5√7 + 25
= 32 - 10√7
Simplify by combining like terms.
Use the FOIL method.
Expand the expression.
Simplify by combining like terms.
Use the FOIL method.
CONFIDENTIAL 26
Rationalizing the DenominatorRationalizing the Denominator
Simplify each quotient.
A) √7 √2
= √7 . (√2) √2 (√2)
= √(14) √4
= √(14) 2
Product Property of Square Roots
Multiply by a form of 1 to get a perfect-square radicand in the denominator.
Simplify the denominator.
CONFIDENTIAL 27
B) √7 √(8n)
= √7 √{4(2n)}
= √7 2√(2n)
= √7 . √(2n) 2√(2n) √(2n)
= √(14n) 2√(2n2)
= √(14n) 2 (2n)
= √(14n) 4n
Simplify the denominator.
Write 8n using a perfect-square factor.
Multiply by a form of 1 to get a perfect-square radicand in the denominator.
Simplify the square root in the denominator.
Product Property of Square Roots
Simplify the denominator.
CONFIDENTIAL 28
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