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Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

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Page 1: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Intermediate Algebra

Exam 4 Material

Radicals, Rational Exponents & Equations

Page 2: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Square Roots

• A square root of a real number “a” is a real number that multiplies by itself to give “a”What is a square root of 9?What is another square root of 9?

• What is the square root of -4 ?Square root of – 4 does not exist in the real number system

• Why is it that square roots of negative numbers do not exist in the real number system?No real number multiplied by itself can give a negative answer

• Every positive real number “a” has two square roots that have equal absolute values, but opposite signsThe two square roots of 16 are:The two square roots of 5 are:

33

ROOT) PRINCIPLE :Root Square (Positive

16 and 16 5 and 5

4 and 4 :simplified

Page 3: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Even Roots (2,4,6,…)

• The even “nth” root of a real number “a” is a real number that multiplies by itself “n” times to give “a”

• Even roots of negative numbers do not exist in the real number system, because no real number multiplied by itself an even number of times can give a negative number

• Every positive real number “a” has two even roots that have equal absolute values, but opposite signsThe fourth roots of 16: The fourth roots of 7:

ROOT) PRINCIPLE :RootEven (Positive

44 7- and 72 and 2 :simplified

existnot does 164

44 16 and 16

Page 4: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Radical Expressions

• On the previous slides we have used symbols of the form:

• This is called a radical expression and the parts of the expression are named:

Index:

Radical Sign :

Radicand:

• Example:

n a

n

a

5 8 8:Radicand 5:Index

Page 5: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Cube Roots

• The cube root of a real number “a” is a real number that multiplies by itself 3 times to give “a”

• Every real number “a” has exactly one cube root that is positive when “a” is positive, and negative when “a” is negative

Only cube root of – 8:

Only cube root of 6:

23 6

root! cube principle a as such thing No

3 8

Page 6: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Odd Roots (3,5,7,…)

• The odd nth root of a real number “a” is a real number that multiplies by itself “n” times to give “a”

• Every real number “a” has exactly one odd root that is positive when “a” is positive, and negative when “a” is negativeThe only fifth root of - 32:

The only fifth root of -7:

3 32

5 7

2

Page 7: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Rational, Irrational, and Non-real Radical Expressions

• is non-real only if the radicand is negative and the index is even

• represents a rational number only if the radicand can be written as a “perfect nth” power of an integer or the ratio of two integers

• represents an irrational number only if it is a real number and the radicand can not be written as “perfect nth” power of an integer or the ratio of two integers

.

n a

5

232 because rational is 325

n a

n a2325

even isindex and negative is radicand because real-non is 206

integers twoof ratio or theinteger an of

powerfourth not the is 8 because irrational is 84

4 8

Page 8: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.1

• Page: 666

• Problems: All: 1 – 6, Odd: 7 – 31, 39 – 57, 65 – 91

• MyMathLab Homework Assignment 10.1 for practice

• MyMathLab Quiz 10.1 for grade

Page 9: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Exponential Expressionsan

“a” is called the base“n” is called the exponent

• If “n” is a natural number then “an” means that “a” is to be multiplied by itself “n” times.Example: What is the value of 24 ?(2)(2)(2)(2) = 16

• An exponent applies only to the base (what it touches)Example: What is the value of: - 34 ? - (3)(3)(3)(3) = - 81Example: What is the value of: (- 3)4 ?(- 3)(- 3)(- 3)(- 3) = 81

• Meanings of exponents that are not natural numbers will be discussed in this unit.

Page 10: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Negative Exponents: a-n

• A negative exponent has the meaning: “reciprocate the base and make the exponent positive”

Examples:

.

nn

aa

1

23

3

3

2

9

1

3

12

8

27

2

33

Page 11: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Quotient Rule for Exponential Expressions

• When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent

Examples:

.

nmn

m

aa

a

7

4

5

5

4

12

x

x

374 55

8412 xx

Page 12: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Rational Exponents (a1/n)and Roots

• An exponent of the form

has the meaning: “the nth root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) one”

n

1

aa n ofroot n theis exists,it if , th1

one) (positive) principle theis roots, n twoare there(If1

th na

) give ton times itselfby multiplies (1

aa n

Page 13: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Examples ofRational Exponent of the Form:

1/n

.

2

1

100 10 100) ofroot square (positive

2

1

5 5 5) ofroot square (positive

2

1

3 )exist!not (Does

2

1

3 3 3) ofroot square (negative

4

1

7 7) ofroot fourth (positive4 7

7

1

9 7 9 9) negative ofroot (seventh

6

1

8 )exist!not (Does

Page 14: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Summary Comments about Meaning of a1/n

• When n is odd:– a1/n always exists and is either positive,

negative or zero depending on whether “a” is positive, negative or zero

• When n is even:– a1/n never exists when “a” is negative– a1/n always exists and is positive or zero

depending on whether “a” is positive or zero

Page 15: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Rational Exponents of the Form: m/n

• An exponent of the form m/n has two equivalent meanings:

(1) am/n means find the nth root of “a”, then raise it to the power of “m”

(assuming that the nth root of “a” exists)

(2) am/n means raise “a” to the power of “m” then take the nth root of am (assuming that the nth root of “am” exists)

Page 16: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example of Rational Exponent of the Form: m/n

82/3

by definition number 1 this means find the cube root of 8, then square it:82/3 = 4(cube root of 8 is 2, and 2 squared is 4)

by definition number 2 this means raise 8 to the power of 2 and then cube root that answer:82/3 = 4(8 squared is 64, and the cube root of 64 is 4)

Page 17: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Definitions and Rules for Exponents

• All the rules learned for natural number exponents continue to be true for both positive and negative rational exponents:Product Rule: aman = am+n

Quotient Rule: am/an = am-n

Negative Exponents: a-n = (1/a)n

.

7

2

7

4

33 7

6

3

7

4

7

2

3

37

2

3

7

4

37

4

3

1

Page 18: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Definitions and Rules for Exponents

Power Rules:(am)n = amn

(ab)m = ambm

(a/b)m = am / bm

Zero Exponent: a0 = 1 (a not zero)

.

7

2

7

4

3 49

8

3

7

2

3x 7

2

7

2

3 x

7

2

4

3

7

2

7

2

4

3

0

4

31

Page 19: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

“Slide Rule” for Exponential Expressions

• When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponentExample: Use rule to slide all factors to other part of the fraction:

• This rule applies to all types of exponents• Often used to make all exponents positive

sr

nm

dc

banm

sr

ba

dc

Page 20: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying Products and Quotients Having Factors with

Rational Exponents• All factors containing a common base can be

combined using rules of exponents in such a way that all exponents are positive:

• Use rules of exponents to get rid of parentheses• Simplify top and bottom separately by using product

rules• Use slide rule to move all factors containing a common

base to the same part of the fraction• If any exponents are negative make a final application of

the slide rule

Page 21: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplify the Expression:

6

1

4

31

2

3

1

2

8

yy

yy

6

1

4

31

23

2

2

8

yy

yy

12

2

12

91

3

6

3

2

2

8

yy

yy

12

71

3

8

2

8

y

y

3

8

12

7

1 82

yy

12

32

12

7

16

yy

12

39

16

y

Page 22: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Applying Rules of Exponentsin Multiplying and Factoring

• Multiply:

• Factor out the indicated factor:

2

1

2

1

2

1

2 xxx 2

1

2

1

2

1

2

1

2

1

2

1

22

xxxxxx

2

1

2

1

2

1

2

1

2

1

2

1

22

xxxx 2

1

2

110 22

xxxx

2

1

2

11 221

xxx

4

3

4

1

4

3

;5

xxx

____4

3

x xx

54

3

4

4

4

3

5 xx

Page 23: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Radical Notation

• Roots of real numbers may be indicated by means of either rational exponent notation or radical notation:

n)(expressio RADICAL a called is n a

INDEX thecalled is n

SIGN RADICAL a called is

RADICAND thecalled is a

Page 24: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Notes About Radical Notation

• If no index is shown it is assumed to be 2• When index is 2, the radical is called a “square root”• When index is 3, the radical is called a “cube root”• When index is n, the radical is called an “nth root”• In the real number system, we can only find even

roots of non-negative radicands. There are always two roots when the index is even, but a radical with an even index always means the positive (principle) root

• We can always find an odd root of any real number and the result is positive or negative depending on whether the radicand is positive or negative

Page 25: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Converting Between Radical and Rational Exponent Notation

• An exponential expression with exponent of the form “m/n” can be converted to radical notation with index of “n”, and vice versa, by either of the following formulas:

1.

2.

• These definitions assume that the nth root of “a” exists

n mn

m

aa

3

2

8

3

2

8

mnn

m

aa 42 2

4643 3 28

23 8

Page 26: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Examples

.

7

4

5 47 5

5 98 5

9

8

11

3

4x 11 34 x

7 45 OR

3114 OR x

Page 27: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

.

• If “n” is even, then this notation means principle (positive) root:

• If “n” is odd, then:

• If we assume that “x” is positive (which we often do) then we can say that:

.

n nx

xxn n

xxn n

xxn n

answer) positive insure toneeded value(absolute

Page 28: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.2

• Page: 675

• Problems: All: 1 – 10, Odd: 11 – 47, 51 – 97

• MyMathLab Homework Assignment 10.2 for practice

• MyMathLab Quiz 10.2 for grade

Page 29: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Product Rule for Radicals

• When two radicals are multiplied that have the same index they may be combined as a single radical having that index and radicand equal to the product of the two radicands:

• This rule works both directions:

nnn abba

nnn baab

44 53 4 53 4 15

3 16 33 28 3 22

Page 30: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Quotient Rule for Radicals

• When two radicals are divided that have the same index they may be combined as a single radical having that index and radicand equal to the quotient of the two radicands

• This rule works both directions:

.

nn

n

b

a

b

a

n

n

n

b

a

b

a

3

3

8

5

4

4

7

54

7

5

3

8

5

2

53

Page 31: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Root of a Root Rule for Radicals

• When you take the mth root of the nth root of a radicand “a”, it is the same as taking a single root of “a” using an index of “mn”

.

mnm n aa

4 3 6 12 6

Page 32: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

NO Similar Rules for Sum and Difference of Radicals

.

nnn baba

nnn baba

35827 333 3523 3

19827 333

1923 3

Page 33: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying Radicals

• A radical must be simplified if any of the following conditions exist:

1. Some factor of the radicand has an exponent that is bigger than or equal to the index

2. There is a radical in a denominator (denominator needs to be “rationalized”)

3. The radicand is a fraction4. All of the factors of the radicand have

exponents that share a common factor with the index

Page 34: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying when Radicand has Exponent Too Big

1. Use the product rule to write the single radical as a product of two radicals where the first radicand contains all factors whose exponents match the index and the second radicand contains all other factors

2. Simplify the first radical

3 42

33 3 22

3 22

Page 35: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

3 5224 yx

3 52332 yx

3 223 33 32 yxy

3 2232 yxy

big? tooishat exponent tanother thereIs

Problem?

:radicals twoofproduct a as thisWrite

:radicalfirst heSimplify t

Page 36: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying when a Denominator Contains a Single Radical of

Index “n”1. Simplify the top and bottom separately to get rid of

exponents under the radical that are too big2. Multiply the whole fraction by a special kind of “1”

where 1 is in the form of:

3. Simplify to eliminate the radical in the denominator

n

n

m

m

"n"

m

toequal be radicand in theexponent every make

torequired factors theall ofproduct theis and

Page 37: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

5 634

3

yx 5 6322

3

yx

2

5 42

2

83

xy

yx

5 325 5 2

3

yxy

5 322

3

yxy

5 423

5 423

5 32 2

2

2

3

yx

yx

yxy

5 555

5 423

2

23

yxy

yx

2

5 423

2

23

xy

yx

Page 38: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying when Radicand is a Fraction

1. Use the quotient rule to write the single radical as a quotient of two radicals

2. Use the rules already learned for simplifying when there is a radical in a denominator

Page 39: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

5

4

35

5

4

3

5 2

5

2

3

5 3

5 3

5 2

5

2

2

2

3

5 5

5 3

2

23

2

245

Page 40: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying when All Exponents in Radicand Share a Common

Factor with Index1. Divide out the common factor from the index

and all exponents

Problem?

6 286432 yx

3 43232 yx

factor? what shareindex and radicandin exponents All 2:gives 2by index andin exponents all Dividing

3 23 33 23 xyx 3 43 xyx

Page 41: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying Expressions Involving Products and/or Quotients of Radicals with the Same Index

• Use the product and quotient rules to combine everything under a single radical

• Simplify the single radical by procedures previously discussed

Page 42: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

4 33

44 3

ba

abab4

33

42

ba

ba 4

a

b

4

4

a

b

4 3

4 3

4

4

a

a

a

b

4 4

4 3

a

ba

a

ba4 3

Page 43: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Right Triangle

• A “right triangle” is a triangle that has a 900 angle (where two sides intersect perpendicularly)

• The side opposite the right angle is called the “hypotenuse” and is traditionally identified as side “c”

• The other two sides are called “legs” and are traditionally labeled “a” and “b”

090

hypotenusecba

Page 44: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Pythagorean Theorem

• In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs:

090

cb

a

222 bac

Page 45: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Pythagorean Theorem Example

• It is a known fact that a triangle having shorter sides of lengths 3 and 4, and a longer side of length 5, is a right triangle with hypotenuse 5.

• Note that Pythagorean Theorem is true:

090

53

4

222 bac 222 345

91625

Page 46: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Using the Pythagorean Theorem

• We can use the Pythagorean Theorem to find the third side of a right triangle, when the other two sides are known, by finding, or estimating, the square root of a number

Page 47: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Using the Pythagorean Theorem

• Given two sides of a right triangle with one side unknown:– Plug two known values and one unknown

value into Pythagorean Theorem– Use addition or subtraction to isolate the

“variable squared”– Square root both sides to find the desired

answer

Page 48: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

• Given a right triangle with find the other side.

25 and 7 ca

222 bac 222 725 b249625 b

2494949625 b2576 b

b57624

Page 49: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.3

• Page: 685

• Problems: Odd: 7 – 19, 23 – 57, 61 – 107

• MyMathLab Homework Assignment 10.3 for practice

• MyMathLab Quiz 10.3 for grade

Page 50: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Adding and Subtracting Radicals

• Addition and subtraction of radicals can always be indicated, but can be simplified into a single radical only when the radicals are “like radicals”

• “Like Radicals” are radicals that have exactly the same index and radicand, but may have different coefficientsWhich are like radicals?

• When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients

344 53 and 52- ,54 ,53

44 5253

34 5352 -

4 55

radicals unlike combinet can' - is asOkay

Page 51: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Note Concerning Adding and Subtracting Radicals

• When addition or subtraction of radicals is indicated you must first simplify all radicals because some radicals that do not appear to be like radicals become like radicals when simplified

Page 52: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

333 16225128 3 433 7 22252

33 3333 33 22225222 333 22225222

333 242524 3 23

(yet) termslikeNot :radicals individualSimplify

:radicals like All

Page 53: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.4

• Page: 691

• Problems: Odd: 5 – 57

• MyMathLab Homework Assignment 10.4 for practice

• MyMathLab Quiz 10.4 for grade

Page 54: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying when there is a Single Radical Term in a Denominator

1. Simplify the radical in the denominator

2. If the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special radical” over itself

3. The “special radical” is one that contains the factors necessary to make the denominator radical factors have exponents equal to index

4. Simplify radical in denominator to eliminate it

Page 55: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

3

3

9

2

x

3 2

3

3

2

x

3 2

3 2

3 2

3

3

3

3

2

x

x

x

3 33

3 2

3

32

x

x

x

x

3

63 21:rdenominatoSimplify

:"1" specialby Multiply

:ruleproduct Use

:rdenominatoSimplify

Page 56: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Simplifying to Get Rid of a Binomial Denominator that Contains One or

Two Square Root Radicals1. Simplify the radical(s) in the denominator2. If the denominator still contains a radical,

multiply the fraction by “1” where “1” is in the form of a “special binomial radical” over itself

3. The “special binomial radical” is the conjugate of the denominator (same terms – opposite sign)

4. Complete multiplication (the denominator will contain no radical)

Page 57: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example

23

5

gsimplifyin needt doesn'r denominatoin Radical

:one specialby fraction Multiply

23

23

23

5

:on top Distribute

:bottomon FOIL

49

1015

:bottomSimplify

23

1015

1015

Page 58: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.5

• Page: 700

• Problems: Odd: 7 – 105

• MyMathLab Homework Assignment 10.5 for practice

• MyMathLab Quiz 10.5 for grade

Page 59: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Radical Equations

• An equation is called a radical equation if it contains a variable in a radicand

• Examples:

53 xx

024 33 xx

15 xx

Page 60: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Solving Radical Equations

1. Isolate ONE radical on one side of the equal sign

2. Raise both sides of equation to power necessary to eliminate the isolated radical

3. Solve the resulting equation to find “apparent solutions”

4. Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, BUT if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions

Page 61: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Why Check When Both Sides are Raised to an Even Power?

• Raising both sides of an equation to a power does not always result in equivalent equations

• If both sides of equation are raised to an odd power, then resulting equations are equivalent

• If both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced)

• Raising both sides to an even power, may make a false statement true:

• Raising both sides to an odd power never makes a false statement true:

.

etc. ,22- ,22- :however , 22 4422

etc. ,22- ,22- :and , 22 5533

Page 62: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example of SolvingRadical Equation

53 xx

35 xx

22 35 xx

325102 xxx

028112 xx 074 xx

07 OR 04 xx7 OR 4 xx

4xCheck

?5344 ?514

53

7xCheck

?5377 ?547

55

solution a NOT is 4x

solution a IS 7x

Page 63: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example of SolvingRadical Equation

15 xxxx 15

2215 xx

xxx 215

x24 x 2

222 x

x4

4xCheck

?1544 ?194

?132 15

solution a NOT is 4x

Solution! No hasEquation

Page 64: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Example of SolvingRadical Equation

024 33 xx33 24 xx

33

33 24 xx

xx 24 x4

check) toneed (No

Page 65: Intermediate Algebra Exam 4 Material Radicals, Rational Exponents & Equations

Homework Problems

• Section: 10.6

• Page: 709

• Problems: Odd: 7 – 57

• MyMathLab Homework Assignment 10.6 for practice

• MyMathLab Quiz 10.6 for grade