CONFIDENTIAL 1

Algebra1Algebra1

Multiplying and Multiplying and DividingDividing

Radical ExpressionsRadical Expressions

CONFIDENTIAL 2

Warm UpWarm Up

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!

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!

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)

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

CONFIDENTIAL 17

Assessment

1) √2√3

2) √3√8

3) (5√2)2

4) 3√(3a)√(10)

5) 2√(15p)√(3p)

Multiply. Write each product in simplest form.

CONFIDENTIAL 18

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)(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

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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