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Algebra1 Multiplying and Dividing Radical Expressions

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.

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Algebra1 Multiplying and Dividing Radical Expressions

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  1. Algebra1Multiplying and DividingRadical Expressions CONFIDENTIAL

  2. Warm Up Simplify. All variables represent nonnegative numbers. 1) √(360) 2) √(72) √(16) 3) √(49x2) √(64y4) 4) √(50a7) √(9a3) 1) 6√(10) 2) 3√2 2 3) 7x 8y2 4) 5a2√2 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 Product Property of Square Roots Multiply the factors in the radicand. 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 Expand the expression. Commutative Property of Multiplication 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) 1a) 5√2 1b) 63 1c) 2m√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 Distribute √2. Product Property of Square Roots. Multiply the factors in the second radicand. Factor 24 using a perfect-square factor. Product Property of Square Roots Simplify. CONFIDENTIAL

  8. B) √3(√3 - √5) = √3.√3- √3.√5 = √{3.(3)} - √{3.(5)} = √9 - √(15) = 3 - √(15) Distribute √3. Product Property of Square Roots. 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) 2a) 4√3 - 3√6 2b) 5√2 + 4√(15) 2c) 7√k - 5√(7)k 2d) 150 - 20√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 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 Use the FOIL method. Simplify by combining like terms. Expand the expression. Use the FOIL method. Simplify by combining like terms. CONFIDENTIAL

  12. Now you try! Multiply. Write each product in simplest form. 3a) (3 + √3)(8 - √3) 3b) (9 + √2)2 3c) (3 - √2)2 3d) (4 - √3)(√3 + 5) 3a) 21 + 5√3 3b) 83 + 18√2 3c) 11 - 6√2 3d) 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 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. Product Property of Square Roots 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 denominator. Multiply by a form of 1 to get a perfect-square radicand in the denominator. Product Property of Square Roots Simplify the square root in the denominator. Simplify the denominator. CONFIDENTIAL

  16. Now you try! Simplify each quotient. 4b) √(7a) √(12) 4a) √(13) √5 4c) 2√(80) √7 4a) √(65) 5 4b) √(21a) 6 4c) 8√(35) 7 CONFIDENTIAL

  17. Assessment Multiply. Write each product in simplest form. • √2√3 • √3√8 • (5√2)2 • 3√(3a)√(10) • 2√(15p)√(3p) 1) √6 2) 2√6 3) 125 4) 3√(30a) 5) 6p√5 CONFIDENTIAL

  18. Multiply. Write each product in simplest form. 6) 2√6 + √(42) 7) 5√3 - 3 8) √(35) - √(21) 9) 2√5 + 16 10) 5√(3y) + 4√(5y) • √6(2 + √7) • √3(5 - √3) • √7{√5 - √3) • √2{√(10) - 8√2} • √(5y){√(15) + 4} CONFIDENTIAL

  19. Multiply. Write each product in simplest form. • (2 + √2) (5 + √2) • (4 + √6) (3 - √6) • (√3 - 4) (√3 + 2) • (5 + √3)2 • (√6 - 5√3)2 11) 12 - 7√2 12) 6 - √6 13) -5√ - 2√3 14) 28 + 10√3 15) 54 + 36√2 CONFIDENTIAL

  20. Simplify each quotient. 17) √(11) 6√3 16) √(20) √8 19) √3 √6 18) √(28) √(3s) 16) √(10) √2 17) √(33) 18 18) 2√(21s) 3s 19) √6 2 20) √(3x) x 20) √3 √x CONFIDENTIAL

  21. Let’s review Multiplying Square Roots Multiply. Write each product in simplest form. A) √3√6 = √{(3)6} = √(18) = √{(9)2} = √9√2 = 3√2 Product Property of Square Roots Multiply the factors in the radicand. Factor 18 using a perfect-square factor. Product Property of Square Roots Simplify. CONFIDENTIAL

  22. B) (5√3)2 = (5√3)(5√3) = 5(5).√3√3 = 25√{(3)3} = 25√9 = 25(3) = 75 Expand the expression. Commutative Property of Multiplication Product Property of Square Roots Simplify the radicand. Simplify the square root. Multiply. 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 Distribute √2. Product Property of Square Roots. Multiply the factors in the second radicand. Factor 24 using a perfect-square factor. Product Property of Square Roots 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 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 Use the FOIL method. Simplify by combining like terms. Expand the expression. Use the FOIL method. Simplify by combining like terms. CONFIDENTIAL

  26. Rationalizing 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. Product Property of Square Roots 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 Write 8n using a perfect-square factor. Simplify the denominator. Multiply by a form of 1 to get a perfect-square radicand in the denominator. Product Property of Square Roots Simplify the square root in the denominator. Simplify the denominator. CONFIDENTIAL

  28. You did a great job today! CONFIDENTIAL

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