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Section R.1 Sets R-1

Copyright © 2017 Pearson Education, Inc.

Chapter R Review of Basic Concepts R.1 Sets ■ Basic Definitions ■ Operations on Sets Key Terms: set, elements (members), infinite set, finite set, Venn diagram, disjoint sets Basic Definitions CLASSROOM EXAMPLE 1 Using Set Notation and Terminology Identify each set as finite or infinite. Then determine whether 8 is an element of the set.

(a) 5, 6, 7, ..., 10 (b) 1 1 1

1, , , , ...2 3 4

(c) | is a fraction between 9 and 10x x

(d) | is a natural number between 7 and 9x x

R-2 Chapter R Review of Basic Concepts


CLASSROOM EXAMPLE 2 Listing the Elements of a Set Use set notation, and list the elements belonging to each set.

(a) | is a natural number between 8 and 12x x

(b) | is a natural number greater than 6 and less than 8x x

CLASSROOM EXAMPLE 3 Examining Subset Relationships Let 3, 9, 15, 21, 27, 33, 39 , 3, 9, 15, 21, 27, 33 , 3, 9, 21, 27 ,U A B

9, 27, 33 ,C and 3, 27 .D Determine whether each statement is true or false.

(a) B D (b) D B (c) D A (d) U A

Section R.1 Sets R-3


Operations on Sets CLASSROOM EXAMPLE 4 Finding Complements of Set Let {1, 2, 3, 4, 5, 6, 7, 8, 9}, {2, 4, 6, 8},U A and {3, 6, 9}.B Find each set.

(a) A (b) B (c) U (d) CLASSROOM EXAMPLE 5 Finding Intersections of Two Sets Find each of the following. Identify any disjoint sets.

(a) 15, 20, 25, 30 {12, 18, 24, 30}

(b) 3, 6, 9, 12, 15, 18 6, 12, 18, 24 (c) 6, 7, 8 678



CLASSROOM EXAMPLE 6 Finding Unions of Two Sets Find each of the following.

(a) 1, 3, 5, 7, 9 3, 6, 9, 12

(b) 9, 10, 11, 12 10, 12, 14, 16

(c) 2, 6, 10, 14, ... 4, 8, 12, 16,

Set Operations Let A and B be sets, with universal set U.

The complement of set A is the set A′ of all elements in the universal set that _________ belong to set A.

{ }′ | , ____A x x U x A= ∈

The intersection of sets A and B, written ,A B is made up of all the elements belonging to both set A ______ set B.

| ____A B x x A x B= ∈ ∈

The union of sets A and B, written ,A B is made up of all the elements belonging to set A ______ set B.

| ____A B x x A x B= ∈ ∈

Section R.2 Real Numbers and Their Properties R-5


R.2 Real Numbers and Their Properties ■ Sets of Numbers and the Number Line ■ Exponents ■ Order of Operations ■ Properties of Real Numbers ■ Order on the Number Line ■ Absolute Value Key Terms: number line, coordinate, coordinate system, algebraic expression, exponential expression (exponential), base, exponent, absolute value Sets of Numbers and the Number Line Sets of Numbers

Set Description

Natural Numbers {1, 2, 3, 4, ...}

Whole Numbers {0, 1, 2, 3, 4, ...}

Integers {..., 3, 2, 1, 0, 1, 2, 3, ...}− − −

Rational numbers and are integers and 0p p q qq

≠

Irrational numbers { | is real but not rational}x x

Real numbers { | corresponds to a point on a number line}x x

CLASSROOM EXAMPLE 1 Identifying Sets of Numbers

Let 5 21

12, 7.25, 3, , 0, 0.45, 9, , 999 .6 3

S = − − − −

List the elements from A that belong

to each set.

(a) Natural numbers (b) Whole numbers (c) Integers (d) Rational numbers (e) Irrational numbers (f) Real numbers



Exponents Exponential Notation

If n is any positive integer and a is any real number, then the nth power of a is written using exponential notation as follows.

_____ factors of

...

a

= na a a a a

CLASSROOM EXAMPLE 2 Evaluating Exponential Expressions Evaluate each exponential expression, and identify the base and the exponent.

(a) 310 (b) 4( 3) (c) 43 (d) 22 5 (e) 2(2 5)



Order of Operations Order of Operations

If grouping symbols such as parentheses, square brackets, absolute value bars, or fraction bars are present, begin as follows.

Step 1 Work separately above and below each fraction bar.

Step 2 Use the rules below within each set of parentheses or square brackets. Start with the innermost set and work outward.

If no grouping symbols are present, follow these steps.

Step 1 Simplify all powers and ____________. Work from left to right.

Step 2 Do any multiplications or ____________ in order. Work from left to right.

Step 3 Do any negations, additions, or ____________ in order. Work from left to right. CLASSROOM EXAMPLE 3 Using Order of Operations Evaluate each expression.

(a) 53 9 2 4 − ÷ (b) (30 5) 3 15 7− ÷ +

(c) 42 11

9 3 2−

− (d)

27 ( 9)

6( 3) 1( 2)

− − −− − −



CLASSROOM EXAMPLE 4 Using Order of Operations Evaluate each expression for 4, 3,x y and 6.z

(a) 26 5 3x y z

(b) 24 3( 1)

9

y xz

(c) 4 3

2 2

x y

z x



Properties of Real Numbers Properties of Real Numbers

Let a, b, and c represent real numbers.

Property Description

Closure Properties

a b+ is a real number ab is a real number

The sum or product of two real numbers is a ______________.

Commutative Properties

a b = b + a

ab = ba

+

The sum or product of two real numbers is the ______________ regardless of their order.

Associative Properties

( ) ( )

( ) ( )

a b + c = a + b + c

ab c = a bc

+

The sum or product of three real numbers is the ______________ no matter which two are added or multiplied first.

Identity Properties

There exists a unique real number 0 such that

0a + = a and 0 + a = a.

There exists a unique real number 1 such that

1a = a and 1a = a.

The sum of a real number and ______________ is that real number, and the product of a real number and ______________ is that real number.

Inverse Properties

There exists a unique real number a− such that

( ) 0a + a =− and 0a + a = .−

If 0,a ≠ there exists a unique real number 1a such that

1

1a =a

and 1

1a = .a

The sum of any real number and its negative is ______________, and the product of any nonzero real number and its reciprocal is ______________.

Distributive Properties

( )

( )

a b c = ab + ac

a b c = ab ac

+− −

The product of a real number and the sum (or difference) of two real numbers ______________ the sum (or difference) of the products of the first number and each of the other numbers.



Multiplication Property of Zero

0 0 0 a a

The product of a real number and ______________ is 0.

With the ______________ properties, the order changes, but with the ______________ properties, the grouping changes. CLASSROOM EXAMPLE 5 Simplifying Expressions Use the commutative and associative properties to simplify each expression.

(a) 12 2 18x (b) 435

7t

(c) 4

549

s

CLASSROOM EXAMPLE 6 Using the Distributive Property Rewrite each expression using the distributive property and simplify, if possible.

(a) 8 2m n (b) 3 5r s

(c) 22 55t (d) 3 5 1

284 6 2

p q



Order on the Number Line If the real number a is to the left of the real number b on a number line, then

a is less than b, written ______________.

If the real number a is to the right of b, then

a is greater than b, written ______________. Absolute Value The distance on the number line from a number to 0 is called the absolute value of that number. Because distance cannot be negative, the absolute value of a number is always _________ or 0. Absolute Value

Let a represent a real number.

if 0

if 0

a aa

a a

≥− <

That is, the absolute value of a positive number or 0 equals that ______________, while the absolute value of a negative number equals its ______________ (or opposite). CLASSROOM EXAMPLE 7 Evaluating Absolute Values Evaluate each expression.

(a) 6.85 (b) 50

(c) 2

3 (d) ,y for 2y



CLASSROOM EXAMPLE 8 Measuring Blood Pressure Difference Systolic blood pressure is the maximum pressure produced by each heartbeat. Both low blood pressure and high blood pressure may be cause for medical concern. Therefore, health care professionals are interested in a patient's "pressure difference from normal," or .dP

If 120 is considered a normal systolic pressure, then

120 ,dP P where P is the patient's recorded systolic pressure.

Find dP for a patient with a systolic pressure, P, of 146.

Properties of Absolute Value

Let a and b represent real numbers.

Property Description

1. a ≥ 0 The absolute value of a real number is positive or ________.

2. a a− = The absolute values of a real number and its opposite are ________.

3. a b ab= The product of the absolute values of two real numbers ________ the absolute value of their product.

4. ( 0)a a

bb b

= The quotient of the absolute values of two real numbers ________ the absolute value of their quotient.

5. a + b a b

(the triangle inequality)

The absolute value of the sum of two real numbers is________ ________ or ________ to the sum of their absolute values.



CLASSROOM EXAMPLE 9 Evaluating Absolute Value Expressions Let 13m and 9.n Evaluate each expression.

(a) 3 5m n (b) 2 3m n

m n

Distance between Points on a Number Line

If P and Q are points on a number line with coordinates a and b, respectively, the distance ( , )d P Q between them is given by the following.

( , )d P Q b a= − or ( , )d P Q a b= −

That is, the distance between two points on a number line is the _______________ _______________of the difference between their coordinates in either order. CLASSROOM EXAMPLE 10 Finding the Distance between Two Points Find the distance between –8 and 14.



R.3 Polynomials ■ Rules for Exponents ■ Polynomials ■ Addition and Subtraction ■ Multiplication ■ Division Key Terms: term, numerical coefficient (coefficient), like terms, polynomial, polynomial in x, degree of a term, degree of a polynomial, trinomial, binomial, monomial, descending order, FOIL method Rules For Exponents Rules for Exponents

For all positive integers m and n and all real numbers a and b, the following rules hold.

Rule Example Description

Product Rule m n m+ na a a =

2 3

2 3

5

2 2 2 2 2 2 2

2

2

When multiplying powers of like bases, keep the base and ______________ the exponents.

Power Rule 1

( )m n mna a=

35 5 5 5

5 5 5

5 3

15

4 4 4 4

4

4

4

To raise a power to a power, ______________ the exponents.

Power Rule 2

( )m m mab a b=

3

3 3

7 7 7 7

7 7 7

7

x x x x

x x x

x

To raise a product to a power, raise each ______________ to that power.

Power Rule 3

( 0)

m m

m

a a

b b

b

=

≠

4

4

4

3 3 3 3 3

5 5 5 5 5

3 3 3 3

5 5 5 5

3

5

To raise a quotient to a power, raise the ______________ and ______________ to that power.

Section R.3 Polynomials R-15


CLASSROOM EXAMPLE 1 Using the Product Rule Find each product.

(a) 6 8m m (b) 3 45 6 3r r r

CLASSROOM EXAMPLE 2 Using the Power Rules Simplify. Assume all variables represent nonzero real numbers.

(a) 537 (b) 45 32 y

(c) 53

2

4

z

(d) 23

4

3abc



Zero Exponent

For any nonzero real number a, 0a = ___________. That is, any nonzero number with a zero exponent equals ____________. CLASSROOM EXAMPLE 3 Using the Definition of 0a Evaluate each power.

(a) 08 (b) 08 (c) 08

(d) 08 (e) 083b

Polynomials The product of a number and one or more variables raised to powers is a ____________. The number is the ____________ ____________, or just the ____________, of the variables. ____________ ____________ are terms with the same variables each raised to the same powers.

A ____________ is a term or a finite sum of terms, with only positive or zero integer exponents permitted on the variables. If the terms of a polynomial contain only the variable x, then the polynomial is a ____________ ____________ ____________. The ____________ ____________ ____________ ____________ with one variable is the exponent on the variable. The greatest degree of any term in a polynomial is the ____________ ____________ ____________ ____________.

A polynomial containing exactly three terms is a ____________. A two-term polynomial is a ____________. A single-term polynomial is a ____________.



CLASSROOM EXAMPLE 4 Classifying Expressions as Polynomials Identify each as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.

(a) 5 35 8x x (b) 2 37x xy (c) 8 4yy

Addition and Subtraction CLASSROOM EXAMPLE 5 Adding and Subtracting Polynomials Add or subtract, as indicated.

(a) 3 2 3 217 10 9 10 5x x x x x x

(b) 4 2 4 26 11 21 6 35m m m m



(c) 3 6 6 3 3 6 6 310 5 25 15r s r s r s r s

(d) 2 26 5 3 4 3 2 9z z z z

Multiplication CLASSROOM EXAMPLE 6 Multiplying Polynomials

Multiply 24 5 3 2 7 .t t t



CLASSROOM EXAMPLE 7 Using the FOIL Method to Multiply Two Binomials Find each product.

(a) 7 3 4 5y y

(b) 6 11 6 11p p

(c) 3 2 5 2 5x x x

Special Products

Product of the Sum and Difference of Two Terms Difference of

squares

2 2( )( )x + y x y x y− = −

Square of a Binomial

Perfect squaretrinomials

2 2 2

2 2 2

( ) 2

( ) 2

x + y x + xy + y

x y x xy + y

=− = −



CLASSROOM EXAMPLE 8 Using the Special Products Find each product.

(a) 7 10 7 10m m

(b) 2 24 9 4 9r r

(c) 2 4 2 45 8 5 8x y x y

(d) 28 3z

(e) 235 12z q



CLASSROOM EXAMPLE 9 Multiplying More Complicated Binomials Find each product.

(a) 4 3 7 4 3 7x y x y

(b) 4a b

(c) 34s t



Division CLASSROOM EXAMPLE 10 Dividing Polynomials Divide 3 212 11 5 8n n n by 3 2.n CLASSROOM EXAMPLE 11 Dividing Polynomials with Missing Terms Divide 4 28 12 7 18x x x by 2 2.x

Section R.4 Factoring Polynomials R-23


R.4 Factoring Polynomials ■ Factoring Out the Greatest Common Factor ■ Factoring by Grouping ■ Factoring Trinomials ■ Factoring Binomials ■ Factoring by Substitution Key Terms: factoring, factored form, prime polynomial, factored completely, greatest common factor (GCF) Factoring Out the Greatest Common Factor The process of finding polynomials whose product equals a given polynomial is called _________________. A polynomial with variable terms that cannot be written as a product of two polynomials of lesser degree is a _________________ _________________. A polynomial is _________________ _________________ when it is written as a product of prime polynomials. CLASSROOM EXAMPLE 1 Factoring Out the Greatest Common Factor Factor out the greatest common factor from each polynomial.

(a) 2 46 18a a− (b) 3 2 2 3 2 214 28 21x y x y x y− +

(c) ( ) ( ) ( )3 224 2 16 2 6 2x x x− − − + −



Factoring by Grouping CLASSROOM EXAMPLE 2 Factoring by Grouping Factor each polynomial by grouping.

(a) 2 23 5 15r s r s+ + + (b) 2 2 24 4m n n m n− + − (c) 3 29 15 6 10y y y− + − Factoring Trinomials As shown in the diagram below, factoring is the opposite of multiplication.



CLASSROOM EXAMPLE 3 Factoring Trinomials Factor each trinomial, if possible.

(a) 25 4 12z z+ − (b) 212 5 3t t− − (c) 23 15 16x x− + (d) 224 42 15x x+ +



Factoring Perfect Square Trinomials 2 2

2 2

2 ___________________

2 ___________________

x + xy + y

x xy + y

=− =

CLASSROOM EXAMPLE 4 Factoring Perfect Square Trinomials Factor each trinomial.

(a) 2 249 28 4x xy y+ + (b) 2 281 20 25a b ab Factoring Binomials Factoring Binomials

Difference of Squares 2 2x y− = ________________

Difference of Cubes 3 3x y− = ________________

Sum of Cubes 3 3x + y = ________________

There is no factoring pattern for a ________________ ________________ ________________ in the real number system.



CLASSROOM EXAMPLE 5 Factoring Differences of Squares Factor each polynomial.

(a) 264 49r (b) 6 4169 144u v

(c) 2 22 3 16c d f (d) 2 218 81 25x x y

(e) 2 24 10 25x y y CLASSROOM EXAMPLE 6 Factoring Sums or Differences of Cubes Factor each polynomial.

(a) 3 1000t (b) 3 38r s



(c) 9 12125 216u v Factoring by Substitution CLASSROOM EXAMPLE 7 Factoring by Substitution Factor each polynomial.

(a) 28 3 1 10 3 1 25x x

(b) 33 1 27x

(c) 4 215 6m m

Section R.5 Rational Expressions R-29


R.5 Rational Expressions ■ Rational Expressions ■ Lowest Terms of a Rational Expression ■ Multiplication and Division ■ Addition and Subtraction ■ Complex Fractions Key Terms: rational expression, domain of a rational expression, lowest terms, complex fraction Rational Expressions The quotient of two polynomials P and Q, with 0Q is a ____________________ ____________________. CLASSROOM EXAMPLE 1 Finding the Domain Find the domain of the rational expression.

7 1

3 1

x xx x

Lowest Terms of a Rational Expression Fundamental Principle of Fractions

,( 0 0)ac a

= b cbc b

≠ ≠

CLASSROOM EXAMPLE 2 Writing Rational Expressions in Lowest Terms Write each rational expression in lowest terms.

(a) 2

2

12 30

4 25

x xx

(b) 2

2

8 16

8 2

x xx x



Multiplication and Division Multiplication and Division For fractions a

b and ( 0, 0),cd b d the following hold.

a c ac=

b d bd and ( 0)a c a d

= cb d b c

÷ ≠

That is, to find the __________________ of two fractions, multiply their numerators to find the numerator of the __________________. Then multiply their denominators to find the denominator of the __________________. To divide two fractions, multiply the dividend (the first fraction) by the __________________ of the divisor (the second fraction). CLASSROOM EXAMPLE 3 Multiplying or Dividing Rational Expressions Multiply or divide, as indicated.

(a) 6

2

6 28

7 9z

z

(b) 2

2

4 3 10 2 1

2 3 2 4n n n

n n n



(c) 2 2

2 3 2

5 16 3 30 6

12 2 8

z z z zz z z z

(d) 2

3

1 2 3 6

1 3 3x xy y x

x xy x y

Addition and Subtraction Addition and Subtraction For fractions a

b and ( 0, 0),cd b d the following hold.

a c ad + bc+ =

b d bd and

a c ad bc=

b d bd

−−

That is, to add (or subtract) two fractions in practice, find their least common denominator (LCD) and change each fraction to one with the LCD as denominator. The sum (or difference) of their numerators is the numerator of their sum (or difference), and the LCD is the denominator of their sum (or difference). Finding the Least Common Denominator (LCD)

Step 1 Write each ____________________ as a product of prime factors.

Step 2 Form a product of all the different prime factors. Each factor should have as exponent the ____________________ exponent that appears on that factor.



CLASSROOM EXAMPLE 4 Adding or Subtracting Rational Expressions Add or subtract, as indicated.

(a) 4 2

3 2

10 15z z

(b) 7 2

5 5

mm m

(c) 4 6

3 5 5 5x x x x



Complex Fractions CLASSROOM EXAMPLE 5 Simplifying Complex Fractions Simplify each complex fraction. In part (b), use two methods.

(a) 2

2

43

16

x

x

(b)

1 1

1 11 1

1

z z

z z



R.6 Rational Exponents ■ Negative Exponents and the Quotient Rule ■ Rational Exponents ■ Complex Fractions Revisited Negative Exponents and the Quotient Rule Negative Exponent Suppose that a is a nonzero real number and n is any integer.

1nn

a =a

−

CLASSROOM EXAMPLE 1 Using the Definition of a Negative Exponent Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers.

(a) 310 (b) 15 (c) 2

4

9

(d) 4mn (e) 4mn

A negative exponent indicates ______________, not a sign change of the expression. Quotient Rule Suppose that m and n are integers and a is a nonzero real number.

mm n

n

a= a

a−

That is, when dividing powers of like bases, keep the same _____________ and subtract the _____________ of the denominator from the _____________ of the numerator.

Section R.6 Rational Exponents R-35


CLASSROOM EXAMPLE 2 Using the Quotient Rule Simplify each expression. Assume all variables represent nonzero real numbers.

(a) 8

3

15

15 (b)

4

9

yy

(c) 6

4

35

25

rr (d)

8 11

12 5

34

51

a ba b

CLASSROOM EXAMPLE 3 Using the Rules for Exponents Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers.

(a) 33 1 45 2x x (b)

4 9

6 3

30

45

r sr s

(c)

2 33 1

41 3

4 4

4

b b

b



Rational Exponents The Expression 1/ na

1/ ,na n Even If n is an even positive integer, and if 0,a then 1/ na is the positive

real number whose nth power is a. That is, 1/ .nna a (In this case,

1/ na is the principal nth root of a.)

1/ ,na n Odd If n is an odd positive integer, and a is any nonzero real number, then

1/ na is the positive or negative real number whose nth power is a. That

is, 1/ .nna a

For all positive integers n, 1/0 0.n CLASSROOM EXAMPLE 4 Using the Definition of 1/na Evaluate each expression.

(a) 1/249 (b) 1/ 2144 (c) 1/ 2(144) (d) 1/664



(e) 1/6( 64) (f) 1/6646 (g) 1/3( 125) (h) 1/364 The Expression /m na

Let m be any integer, n be any positive integer, and a be any real number for which 1/na is a real number.

/ 1/ mm n na a=

CLASSROOM EXAMPLE 5 Using the Definition of /m na Evaluate each expression.

(a) 3/481 (b) 3/ 225



(c) 5/ 24 (d) 2/3( 64) (e) 2/3216 (f) 3/ 2( 100) Definitions and Rules for Exponents Suppose that r and s represent rational numbers. The results here are valid for all positive numbers a and b.

Product rule r sa a ______=

Quotient rule r

s

a______

a=

Negative exponent ra ______− =

Power rules ( )r sa ______=

( )rab ______=

ra

______b

=

CLASSROOM EXAMPLE 6 Using the Rules for Exponents Simplify each expression. Assume all variables represent positive real numbers.

(a) 1/ 2 7 / 2

3

18 18

18

(b) 3/ 2 3/ 4100 16



(c) 3/ 4 2/54 5z z (d) 2 1/34/3 4

2/3 5

5

8

m mn n

(e) 3/7 4/7 11/75y y y

CLASSROOM EXAMPLE 7 Factoring Expressions with Negative or Rational Exponents Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers.

(a) 5 228 21y y (b) 4/3 1/318 12n n



(c) 2/5 3/53 3x x

Complex Fractions Revisited CLASSROOM EXAMPLE 8 Simplifying a Fraction with Negative Exponents

Simplify 1 1

2 2.

x yx y

Write the result with only positive exponents.

Section R.7 Radical Expressions R-41


R.7 Radical Expressions ■ Radical Notation ■ Simplified Radicals ■ Operations with Radicals ■ Rationalizing Denominators Key Terms: radicand, principal nth root, like radicals, unlike radicals, conjugates Radical Notation Radical Notation for 1/na Let a be a real number, n be a positive integer, and 1/na be a real number.

1/ nn a = a

Radical Notation for /m na

Let a be a real number, m be an integer, n be a positive integer, and n a be a real number.

( )/m

m n na = a = ________________

CLASSROOM EXAMPLE 1 Evaluating Roots Write each root using exponents and evaluate.

(a) 3 27 (b) 4 10,000

(c) 3 216 (d) 4 81

(e) 3125

512 (f) 5 243



CLASSROOM EXAMPLE 2 Converting from Rational Exponents to Radicals Write in radical form and simplify. Assume all variable expressions represent positive real numbers.

(a) 3/ 416 (b) 2/364

(c) 3/ 2121 (d) 7/87 (e) 4/57z (f) 1/ 412q

(g) 1/65 2x y

CLASSROOM EXAMPLE 3 Converting from Radicals to Rational Exponents Write in exponential form. Assume all variable expressions represent positive real numbers.

(a) 7 3n (b) 4 10x (c) 4315 r



(d) 8252 3x (e) 3 2 4r s

Evaluating n na

If n is an even positive integer, then .n na a=

If n is an odd positive integer, then .n na a=

CLASSROOM EXAMPLE 4 Using Absolute Value to Simplify Roots Simplify.

(a) 6z (b) 7 7t

(c) 8 1081r s (d) 44 ( 3)

(e) 5 10m (f) 23 4x



(g) 2 10 25x x Rules for Radicals Suppose that a and b represent real numbers, and m and n represent positive integers for which the indicated roots are real numbers.

Rule Description

Product Rule

n n na b ab =

The product of two roots is the root of the _______________.

Quotient Rule

( 0)n

nn

a ab

b b= ≠

The root of a quotient is the _______________ of the roots.

Power Rule

m n mna a=

The index of the root of a root is the _______________ of their indexes.

CLASSROOM EXAMPLE 5 Simplifying Radical Expressions Simplify. Assume all variable expressions represent positive real numbers.

(a) 5 45 (b) 5 53 2n n



(c) 11

169 (d) 6

12

ab

(e) 5 4 17 (f) 6 8 Simplified Radicals Simplified Radicals

An expression with radicals is simplified when all of the following conditions are satisfied. 1. The radicand has no factor raised to a power greater than or equal to the index.

2. The radicand has no fractions.

3. No denominator contains a radical.

4. Exponents in the radicand and the index of the radical have greatest common factor 1.

5. All indicated operations have been performed (if possible).



CLASSROOM EXAMPLE 6 Simplifying Radicals Simplify each radical.

(a) 288 (b) 38 125

(c) 3 6 8 10128a b c Operations with Radicals CLASSROOM EXAMPLE 7 Adding and Subtracting Radicals Add or subtract, as indicated. Assume all variables represent positive real numbers.

(a) 14 5 11 5pq pq



(b) 375 12ab b ab

(c) 5 7 8 43 381 24x y x y

CLASSROOM EXAMPLE 8 Simplifying Radicals Simplify each radical. Assume all variables represent positive real numbers.

(a) 10 52 (b) 3 9 18a b (c) 6 3 24



CLASSROOM EXAMPLE 9 Multiplying Radical Expressions Find each product.

(a) 11 17 11 17

(b) 5 32 3 2

Rationalizing Denominators CLASSROOM EXAMPLE 10 Rationalizing Denominators Rationalize each denominator.

(a) 2

7



(b) 34

9

CLASSROOM EXAMPLE 11 Simplifying Radical Expressions with Fractions Simplify each expression. Assume all variables represent positive real numbers.

(a) 3 5

3 2 5

a ba b

(b) 4 48 16

6 3

x x

CLASSROOM EXAMPLE 12 Rationalizing a Binomial Denominator Rationalize the denominator.

2

3 5

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