algebra1 multiplying and dividing radical expressions

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.


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.


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.

CONFIDENTIAL 6

Now you try!

1a) 5√21b) 631c) 2m√7


1a) √5√(10)

1b) (3√7)2

1c) √(2m) + √(14m)

CONFIDENTIAL 7

Using the Distributive PropertyUsing the Distributive Property


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


Distribute √2.

Multiply the factors in the second radicand.


Simplify.


CONFIDENTIAL 8

B) √3(√3 - √5)

= √3.√3 - √3.√5

= √{3.(3)} - √{3.(5)}

= √9 - √(15)

= 3 - √(15)


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


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


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.




CONFIDENTIAL 12


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


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


Write 8n using a perfect-square factor.


Simplify the square root in the denominator.



CONFIDENTIAL 16


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


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}


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


CONFIDENTIAL 20


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


A) √3√6

= √{(3)6}

= √(18)

= √{(9)2}

= √9√2

= 3√2





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



Simplify the radicand.

Simplify the square root.

Multiply.

CONFIDENTIAL 23

Using the Distributive PropertyUsing the Distributive Property


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


Distribute √2.

Multiply the factors in the second radicand.


Simplify.


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


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






CONFIDENTIAL 26

Rationalizing the DenominatorRationalizing the Denominator


A) √7 √2

= √7 . (√2) √2 (√2)

= √(14) √4

= √(14) 2




CONFIDENTIAL 27

B) √7 √(8n)

= √7 √{4(2n)}

= √7 2√(2n)

= √7 . √(2n) 2√(2n) √(2n)

= √(14n) 2√(2n2)

= √(14n) 2 (2n)

= √(14n) 4n


Write 8n using a perfect-square factor.


Simplify the square root in the denominator.



CONFIDENTIAL 28

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