1. CONFIDENTIAL 1 Algebra1 Multiplying and Dividing Radical Expressions

2. CONFIDENTIAL 2 Warm Up 1) √(360) 2) √(72) √(16) 3) √(49x 2 ) √(64y 4 ) 4) √(50a 7 ) √(9a 3 ) Simplify. All variables represent nonnegative numbers.

3. CONFIDENTIAL 3 Multiplying 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.

4. 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.

5. CONFIDENTIAL 5 C) 2√(8x)√(4x) = 2√{(8x)(4x)} = 2√(32x 2 ) = 2√{(16)(2)(x 2 )} = 2√(16)√2√(x 2 ) = 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.

6. CONFIDENTIAL 6 Now you try! Multiply. Write each product in simplest form. 1a) √5√(10) 1b) (3√7) 2 1c) √(2m) + √(14m)

7. CONFIDENTIAL 7 Using 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

8. 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.

9. 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)

10. 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.

11. CONFIDENTIAL 11 Multiplying Sums and Differences of Radicals 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.

12. 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)

13. 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.

14. CONFIDENTIAL 14 Rationalizing 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.

15. CONFIDENTIAL 15 B) √7 √(8n) = √7 √{4(2n)} = √7 2√(2n) = √7. √(2n) 2√(2n) √(2n) = √(14n) 2√(2n 2 ) = √(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.

16. CONFIDENTIAL 16 Simplify each quotient. Now you try! 4a) √(13) √5 4b) √(7a) √(12) 4c) 2√(80) √7

17. 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.

18. 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.

19. 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.

20. CONFIDENTIAL 20 Simplify each quotient. 16) √(20) √8 17) √(11) 6√3 18) √(28) √(3s) 19) √3 √6 20) √3 √x

21. CONFIDENTIAL 21 Multiplying 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

22. 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.

23. CONFIDENTIAL 23 Using 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

24. 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.

25. CONFIDENTIAL 25 Multiplying Sums and Differences of Radicals 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.

26. CONFIDENTIAL 26 Rationalizing 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.

27. CONFIDENTIAL 27 B) √7 √(8n) = √7 √{4(2n)} = √7 2√(2n) = √7. √(2n) 2√(2n) √(2n) = √(14n) 2√(2n 2 ) = √(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.

28. CONFIDENTIAL 28 You did a great job today!