Review of Basic Concepts

Transcription

1 R Review of Basic Concepts Sections R.5 R.7 Copyright 2013, 2009, 2005 Pearson Education. Inc.

2 R Review of Basic Concepts R.5 Rational Expressions R.6 Rational Exponents R.7 Radical Expressions Copyright 2013, 2009, 2005 Pearson Education. Inc. R-74

3 R.5 Rational Expressions Rational Expressions Lowest Terms of a Rational Expression Multiplication and Division Addition and Subtraction Complex Fractions Copyright 2013, 2009, 2005 Pearson Education. Inc. R-75

4 R.5 Example 1 Finding the Domain (page 41) Find the domain of the rational expression. ( x 7)( x 1) ( x+ 3)( x 1) Set the denominator equal to zero. x+ 3 = 0 or x 1= 0 x = 3 or x = 1 { x x 3,1} Copyright 2013, 2009, 2005 Pearson Education. Inc. R-76

5 R.5 Example 2(a) Writing Rational Expressions in Lowest Terms (page 42) Write the rational expression in lowest terms. (a) 2 12x x 25 x = 6x(2x 5) (3x + 5 )(2x 5) Factor. = 6x 2x + 5 Divide out the common factor. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-77

6 R.5 Example 2(b) Writing Rational Expressions in Lowest Terms (page 42) Write the rational expression in lowest terms. (b) 2 x 8x x 2x = 2 ( x 4) 2 x( 4 x) Factor. = 2 ( x 4) 2 x( x 4) Multiply numerator and denominator by 1. x 4 = 2x ( x 4) 4 x = or 2x 2x Divide out the common factor. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-78

7 R.5 Example 3(a) Multiplying or Dividing Rational Expressions (page 43) Multiply. Multiply. Factor. Divide out common factors, then simplify. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-79

8 R.5 Example 3(b) Multiplying or Dividing Rational Expressions (page 43) Multiply. Factor. Multiply. Divide out common factors, then simplify. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-80

9 R.5 Example 3(c) Multiplying or Dividing Rational Expressions (page 43) Divide. Multiply by the reciprocal of the divisor. Factor. Multiply, then divide out common factors. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-81

10 R.5 Example 3(d) Multiplying or Dividing Rational Expressions (page 43) Multiply. Factor. Multiply, then divide out common factors. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-82

11 R.5 Example 4(a) Adding or Subtracting Rational Expressions (page 44) Add Find the LCD: Copyright 2013, 2009, 2005 Pearson Education. Inc. R-83

12 R.5 Example 4(b) Adding or Subtracting Rational Expressions (page 44) Add Find the LCD: Copyright 2013, 2009, 2005 Pearson Education. Inc. R-84

13 R.5 Example 4(c) Adding or Subtracting Rational Expressions (page 44) Subtract Find the LCD: Copyright 2013, 2009, 2005 Pearson Education. Inc. R-85

14 R.5 Example 5(a) Simplifying Complex Fractions (page 46) Simplify Multiply the numerator and denominator by the LCD of all the fractions, x 2. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-86

15 R.5 Example 5(b) Simplifying Complex Fractions (page 46) Simplify Multiply the numerator and denominator by the LCD of all the fractions, z(z + 1)(z 1). Copyright 2013, 2009, 2005 Pearson Education. Inc. R-87

16 R.6 Rational Exponents Negative Exponents and the Quotient Rule Rational Exponents Complex Fractions Revisited Copyright 2013, 2009, 2005 Pearson Education. Inc. R-88

17 R.6 Example 1 Using the Definition of a Negative Exponent (page 50) Evaluate each expression. (a) (b) (c) (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-89

18 R.6 Example 1 Using the Definition of a Negative Exponent (cont.) Write the expression without negative exponents. (d) (e) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-90

19 R.6 Example 2 Using the Quotient Rule (page 51) Simplify each expression. (a) (b) (c) (d) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-91

20 R.6 Example 3(a) Using Rules for Exponents (page 51) Simplify. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-92

21 R.6 Example 3(b) Using Rules for Exponents (page 51) Simplify. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-93

22 R.6 Example 3(c) Using Rules for Exponents (page 51) Simplify. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-94

23 R.6 Example 4 Using the Definition of a 1/n (page 52) Evaluate each expression. (a) (b) (c) (d) (e) not a real number (f) (g) (h) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-95

24 R.6 Example 5 Using the Definition of a m/n (page 53) Evaluate each expression. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-96

25 R.6 Example 5 Using the Definition of a m/n (cont.) Evaluate each expression. (d) (e) (f) is not a real number because is not a real number. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-97

26 R.6 Example 6 Using the Rules for Exponents (page 54) Simplify each expression. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-98

27 R.6 Example 6 Using the Rules for Exponents (cont.) Simplify each expression. (d) (e) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-99

28 R.6 Example 7 Factoring Expressions with Negative or Rational Exponents (page 54) Factor out the least power of the variable or variable expression. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-100

29 R.6 Example 8 Simplifying a Fraction with Negative Exponents (page 55) Simplify. Write the result with only positive exponents. Definition Add fractions of negative exponent Multiply numerator and denominator by the LCD of the fractions. Factor Divide out the common factor Copyright 2013, 2009, 2005 Pearson Education. Inc. R-101

30 R.7 Radical Expressions Radical Notations Simplified Radicals Operations with Radicals Rationalizing Denominators Copyright 2013, 2009, 2005 Pearson Education. Inc. R-102

31 R.7 Example 1 Evaluating Roots (page 59) Write each root using exponents and evaluate. (a) (b) (c) (d) is not a real number. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-103

32 R.7 Example 1 Evaluating Roots (cont.) Write each root using exponents and evaluate. (e) (f) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-104

33 R.7 Example 2 Converting From Rational Exponents to Radicals (page 60) Write in radical form and simplify. (a) (b) (c) (d) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-105

34 R.7 Example 2 Converting From Rational Exponents to Radicals (cont.) Write in radical form and simplify. (e) (f) (g) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-106

35 R.7 Example 3 Converting From Radicals to Rational Exponents (page 60) Write in exponential form. (a) (b) (c) = 15r 4/3 (d) (e) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-107

36 R.7 Example 4 Using Absolute Value to Simplify Roots (page 61) Simplify each expression. (a) (b) (c) (d) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-108

37 R.7 Example 4 Using Absolute Value to Simplify Roots (cont.) Simplify each expression. (e) (f) (g) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-109

38 R.7 Example 5 Simplifying Radical Expressions (page 62) Simplify each expression. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-110

39 R.7 Example 5 Simplifying Radical Expressions (cont.) Simplify each expression. (d) (e) (f) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-111

40 R.7 Example 6 Simplifying Radicals (page 62) Simplify each radical. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-112

41 R.7 Example 7 Adding and Subtracting Like Radicals (page 63) Add or subtract as indicated. (a) (b) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-113

42 R.7 Example 7 Adding and Subtracting Like Radicals (cont.) Add or subtract as indicated. (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-114

43 R.7 Example 8 Simplifying Radicals (page 64) Simplify each radical. (a) (b) (c) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-115

44 R.7 Example 9(a) Multiplying Radical Expressions (page 64) Find the product. Product of the sum and difference of two terms. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-116

45 R.7 Example 9(b) Multiplying Radical Expressions (page 64) Find the product. Simplify FOIL Copyright 2013, 2009, 2005 Pearson Education. Inc. R-117

46 R.7 Example 10 Rationalizing Denominators (page 65) Rationalize each denominator. (a) (b) Copyright 2013, 2009, 2005 Pearson Education. Inc. R-118

47 R.7 Example 11(a) Simplifying Radical Expressions with Fractions (page 65) Simplify the expression. Simplify Quotient radicand. rule Rationalize denominator. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-119

48 R.7 Example 11(b) Simplifying Radical Expressions with Fractions (page 65) Simplify the expression. Quotient rule Simplify the denominators. Write with a common denominator. Subtract the numerators. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-120

49 R.7 Example 12 Rationalizing a Binomial Denominator (page 66) Rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator. Copyright 2013, 2009, 2005 Pearson Education. Inc. R-121