algebra1 multiplying and dividing radical expressions
DESCRIPTION
Algebra1 Multiplying and Dividing Radical Expressions. Warm Up. Simplify. All variables represent nonnegative numbers. 1) √(360) 2) √(72) √(16) 3) √(49x 2 ) √(64y 4 ) 4) √(50a 7 ) √(9a 3 ). 1) 6√(10) 2) 3 √2 2 3) 7x 8y 2 4) 5a 2 √2 3. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Algebra1 Multiplying and Dividing Radical Expressions](https://reader035.vdocuments.mx/reader035/viewer/2022062217/5681380b550346895d9fc2a7/html5/thumbnails/1.jpg)
CONFIDENTIAL 1
Algebra1Algebra1
Multiplying and Multiplying and DividingDividing
Radical ExpressionsRadical Expressions
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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.
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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.
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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.
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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.
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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)
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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
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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.
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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)
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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.
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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.
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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
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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
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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
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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.
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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
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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.
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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.
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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.
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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.
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CONFIDENTIAL 28
You did a great job You did a great job today!today!