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

Mar 26, 2015

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

R

Copyright © 2013, 2009, 2005 Pearson Education. Inc.

Review of Basic Concepts

Sections R.5–R.7

Page 2: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-2

R.5 Rational Expressions

R.6 Rational Exponents

R.7 Radical Expressions

Review of Basic ConceptsR

Page 3: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-3

Rational ExpressionsR.5Rational Expressions ▪ Lowest Terms of a Rational Expression ▪ Multiplication and Division ▪ Addition and Subtraction ▪ Complex Fractions

Page 4: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-4

Find the domain of the rational expression.

Set the denominator equal to zero.

R.5 Example 1 Finding the Domain (page 41)

( 7)( 1)

( 3)( 1)

x x

x x

3 0 or 1 0x x

3 or 1x x

{ | 3,1}x x

Page 5: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-5

Write the rational expression in lowest terms.

(a)

R.5 Example 2(a) Writing Rational Expressions in Lowest Terms (page 42)

2

2

12 30

4 25

x x

x

Factor.6

(3

(2 5)

(2 55 )) x

x

x

x

Divide out the common factor.

6

2 5

x

x

Page 6: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-6

Write the rational expression in lowest terms.

(b)

R.5 Example 2(b) Writing Rational Expressions in Lowest Terms (page 42)

2

2

8 16

8 2

x x

x x

Factor.

2( )

2 (

4

4 )

x

x x

Multiply numerator and denominator by –1.

2( )

2 (

4

)4x

x

x

Divide out the common factor.

4

2( 4) 4

or 2 2

x

xx x

x x

Page 7: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-7

R.5 Example 3(a) Multiplying or Dividing Rational Expressions (page 43)

Multiply.

Multiply.

Factor.

Divide out common factors, then simplify.

Page 8: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-8

Multiply.

R.5 Example 3(b) Multiplying or Dividing Rational Expressions (page 43)

Factor.

Multiply.

Divide out common factors, then simplify.

Page 9: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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.

Page 10: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-10

R.5 Example 3(d) Multiplying or Dividing Rational Expressions (page 43)

Multiply.

Factor.

Multiply, then divide out common factors.

Page 11: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-11

R.5 Example 4(a) Adding or Subtracting Rational Expressions (page 44)

Add

Find the LCD:

Page 12: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-12

R.5 Example 4(b) Adding or Subtracting Rational Expressions (page 44)

Add

Find the LCD:

Page 13: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-13

R.5 Example 4(c) Adding or Subtracting Rational Expressions (page 44)

Subtract

Find the LCD:

Page 14: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-14

R.5 Example 5(a) Simplifying Complex Fractions (page 46)

Simplify

Multiply the numerator and denominator by the LCD of all the fractions, x2.

Page 15: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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).

Page 16: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-16

Rational ExponentsR.6Negative Exponents and the Quotient Rule ▪ Rational Exponents ▪ Complex Fractions Revisited

Page 17: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-17

R.6 Example 1 Using the Definition of a Negative Exponent (page 50)

Evaluate each expression.

(a) (b) (c)

(a) (b)

(c)

Page 18: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-18

R.6 Example 1 Using the Definition of a Negative Exponent (cont.)

Write the expression without negative exponents.

(d)

(e)

Page 19: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-19

R.6 Example 2 Using the Quotient Rule (page 51)

Simplify each expression.

(a) (b)

(c)

(d)

Page 20: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-20

R.6 Example 3(a) Using Rules for Exponents (page 51)

Simplify.

Page 21: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-21

R.6 Example 3(b) Using Rules for Exponents (page 51)

Simplify.

Page 22: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-22

R.6 Example 3(c) Using Rules for Exponents (page 51)

Simplify.

Page 23: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-23

R.6 Example 4 Using the Definition of a1/n (page 52)

Evaluate each expression.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

not a real number

Page 24: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-24

R.6 Example 5 Using the Definition of am/n (page 53)

Evaluate each expression.

(a)

(b)

(c)

Page 25: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-25

R.6 Example 5 Using the Definition of am/n (cont.)

Evaluate each expression.

(d)

(e)

(f) is not a real number because is not a real number.

Page 26: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-26

R.6 Example 6 Using the Rules for Exponents (page 54)

Simplify each expression.

(a)

(b)

(c)

Page 27: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-27

R.6 Example 6 Using the Rules for Exponents (cont.)

Simplify each expression.

(d)

(e)

Page 28: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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)

Page 29: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-29

R.6 Example 8 Simplifying a Fraction with Negative Exponents (page 55)

Simplify. Write the result with only positive exponents.

Definition of negative exponentAdd fractions

Multiply numerator and denominator by the LCD of the fractions.

FactorDivide out the common factor

Page 30: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-30

Radical ExpressionsR.7Radical Notations ▪ Simplified Radicals ▪ Operations with Radicals ▪ Rationalizing Denominators

Page 31: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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.

Page 32: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-32

R.7 Example 1 Evaluating Roots (cont.)

Write each root using exponents and evaluate.

(e)

(f)

Page 33: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-33

R.7 Example 2 Converting From Rational Exponents to Radicals (page 60)

Write in radical form and simplify.

(a)

(b)

(c)

(d)

Page 34: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-34

R.7 Example 2 Converting From Rational Exponents to Radicals (cont.)

Write in radical form and simplify.

(e)

(f)

(g)

Page 35: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-35

R.7 Example 3 Converting From Radicals to Rational Exponents (page 60)

Write in exponential form.

(a) (b)

(c)

(d)

(e)

4/315r

Page 36: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-36

R.7 Example 4 Using Absolute Value to Simplify Roots (page 61)

Simplify each expression.

(a)

(b)

(c)

(d)

Page 37: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-37

R.7 Example 4 Using Absolute Value to Simplify Roots (cont.)

Simplify each expression.

(e)

(f)

(g)

Page 38: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-38

R.7 Example 5 Simplifying Radical Expressions (page 62)

Simplify each expression.

(a)

(b)

(c)

Page 39: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-39

R.7 Example 5 Simplifying Radical Expressions (cont.)

Simplify each expression.

(d)

(e)

(f)

Page 40: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-40

R.7 Example 6 Simplifying Radicals (page 62)

Simplify each radical.

(a)

(b)

(c)

Page 41: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-41

R.7 Example 7 Adding and Subtracting Like Radicals (page 63)

Add or subtract as indicated.

(a)

(b)

Page 42: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-42

R.7 Example 7 Adding and Subtracting Like Radicals (cont.)

Add or subtract as indicated.

(c)

Page 43: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-43

R.7 Example 8 Simplifying Radicals (page 64)

Simplify each radical.

(a)

(b)

(c)

Page 44: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-44

R.7 Example 9(a) Multiplying Radical Expressions

(page 64)

Find the product.Product of the sum and difference of two terms.

Page 45: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-45

R.7 Example 9(b) Multiplying Radical Expressions

(page 64)

Find the product.

Simplify

FOIL

Page 46: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-46

R.7 Example 10 Rationalizing Denominators (page 65)

Rationalize each denominator.

(a)

(b)

Page 47: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-47

R.7 Example 11(a) Simplifying Radical Expressions with Fractions (page 65)

Simplify the expression.

Quotient ruleSimplify radicand.

Rationalize denominator.

Quotient rule

Page 48: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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.

Page 49: R Copyright © 2013, 2009, 2005 Pearson Education. Inc. Review of Basic Concepts Sections R.5–R.7.

Copyright © 2013, 2009, 2005 Pearson Education. Inc. R-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.