Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Algebra 2 Multiplying, Dividing, Rationalizing and Simplifying… Section 7-2 Multiplying and Dividing Radical Expressions Lesson 7-2 Objectives Algebra 2 • To multiply radical expressions • To simplify radical expressions • To multiply and simplify radical expressions • To find roots of quotients • To divide radical expressions • Solve algebraic equations involving radical expressions Multiplying and Dividing Radical Expressions Lesson 7-2 Objectives Algebra 2 • To multiply radical expressions • To simplify radical expressions • To multiply and simplify radical expressions • To find roots of quotients • To divide radical expressions • Solve algebraic equations involving radical expressions Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Property For any nonnegative numbers a and b, and any natural-number index k , k a k b k ab. Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Multiply x2 3 4 19 3 x2 x 4 2 5 20 3 7 133 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Factor and Simplify Radical Expressions k ab a k k b Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Simplify 20 4 5 2 5 50 25 2 5 2 3 32 8 4 2 4 3 3 3 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Simplify 2x 4x 2 2 2 x 2 x 1 2 ( x 1) 2 2 x 1 2 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples Multiply. Simplify if possible. a. b. c. 3• 12 3• 12 = 3 –16 • 3 4 3 –16 • 3 4= –4 • 16 3 • 12 = 3 36 = 6 –16 • 4 = 3 64 = –4 The property for multiplying radicals does not apply. –4 is not a real number. Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples Simplify each expression. Assume all variables are positive. 50x5 a. 50x5 = = 52 • 2 • (x2)2 • x Factor into perfect squares. 52 • (x2)2 • n = 5x2 b. 2•x 2x 54n8 3 54n8 = 3 33 • 2 • (n2)3 • n2 = 3 33(n2)3 • 3 2n2 n b = n ab definition of nth root 3 = 3n2 a• 3 2n2 Factor into perfect cubes. n a• n b = n ab definition of nth root Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples Multiply and simplify are positive. 3 25xy8 • 3 3 25xy8 • 5x4y3 = 3 53x3(y3)3 • x2y2 = 3 53x3(y3)3 • = 5xy3 3 x2y2 25xy8 • n 5x4y3 = 3 3 x2y2 a• 3 n 5x4y3 . Assume all variables b = n ab Factor into perfect cubes. n a• n b = n ab definition of nth root Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Property For any natural number index k and any real numbers a and b, (b 0), k k where a and b are real numbers, k a a k k b b. To find the k th root of a quotient, find the k th root of the numerator and denominator separately. Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Dividing Radical Expressions To divide radical expressions with the same index, divide the radicands. Simplify if possible. k k a k a b b Multiplying and Dividing Radical Expressions Lesson 7-2 Dividing Variables w/ Exponents Algebra 2 • Reminder: When Multiplying Expressions with similar Bases the Exponents ADD! • When Dividing Expressions with similar Bases the Exponents SUBTRACT! 3 3 5 32 2 3 4 3 2 16 3 3 1 2 2 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples Divide and simplify. Assume all variables are positive. a. 3 = = –81 3 3 3 b. 3 = –81 3 = 192x8 3 3 3x 192x8 3x = 3 –27 = 3 64x7 = 3 (–3)3 = 3 43(x2)3 • x = –3 = 4x2 3 x Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 27 3 125 3 16x 4 y 3 5 3 27 3 125 16 x y 2 2 y x 2 4x y x 2 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 5 2 6 64a b 5 7 2a b 2 6 5 64a b 7 2a b 1 5 5 5 5 32 5 b a 2 64 a 5 5 7 2 a 6 5 b b 1 2b 2 b a a Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Rationalizing the Denominator Writing the result without radicals in the denominator. Multiply by a 1 to make the denominator a perfect power. Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples Rationalize the denominator of each expression. Assume that all variables are positive. Method 1: a. 3 3 5 5 = 3 5 Rewrite as a square root of a fraction. = 3•5 5•5 Then make the denominator a perfect square. = 15 52 = 15 5 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples (continued) Method 2: a. 3 3 5 5 = 3• 5• = 15 5 5 5 Multiply the numerator and denominator by 5 so the denominator becomes a whole number. Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 1 2 3 6 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 3 3 x 1 3 3 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Additional Examples (continued) b. c. x5 3 5 4y 3 5 4y 3x2y x5 3x2y = = = = x5 • 3x2y 3x2y • 3x2y 3x7y 3x2y x3 x 3xy 2 3x y 3xy 3y = 3 5 • 42y2 4y • 42y2 = 3 80y2 = = 43y3 2 3 10y2 4y 3 10y2 2y Rewrite the fraction so the denominator is a perfect cube. Multiplying and Dividing Radical Expressions Lesson 7-2 Additional Examples Algebra 2 The distance d in meters that an object will fall in t seconds is given by d = 4.9t 2. Express t in terms of d and rationalize the denominator. d = 4.9t 2 d t 2 = 4.9 t= = = d 4.9 d • 10 49 10d 7 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Multiply and Simplify 15 6 90 9 10 3 10 3 25 2 5 6 125 6 5 30 3 x 1 3 x 1 3 ( x 1) x 1 2 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Multiply and Simplify 3 x 2 3 4 x ( x 5) 3 8x 2 3x y 3 4 2x 2 2 8x y 4 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Multiply and Simplify 3 18 y 3 3 4x 2 72 x y 72 x y 2 3 3 3 3 2 3 3 8 9 x y 2 y 9x 3 3 3 2 3 2 Multiplying and Dividing Radical Expressions Lesson 7-2 Algebra 2 Do I have to Multiply and Simplify…Can’t I just Factor then Simplify? 18 y 14 y 252y 9 2 7 2 y 2 2 18 y 14 y 9 2 7 2 y 9 4 7 y 3 2 7 x 2 2 x 2 3 4 5 6 7 8 9 10 x2 4 9 16 25 36 49 64 81 100 x3 8 27 64 125 216 343 512 729 1000 x4 16 81 256 625 x5 Algebra 2 32 243 Algebra 2 Homework Page Simplify Algebra 2 Multiply and Simplify Algebra 2 3 3 0.3x 0.09 x 0.027x 20 5 4 2 5 3 3 54x 3 2 8 27 2 x x x 3 2 23 2 3 3 2 x x x 3x 2 x 3 3 3 3 3 3 3 2 Algebra 2 22. 4 80 16 5 2 5 4 4 4 Algebra 2 25. 3 (x+y) 4 (x+y) 3 (x+y) ( x y ) 3 (x+y) 3 3 Algebra 2 28. 5 10 50 25 2 5 2 Algebra 2 31. 3 2x y 12 xy 24x y 4 6 x y 4 2 4 2 6 x y 2x y 6 2 2 2