Radicals 07/27/12lntaylor ©. Table of Contents Learning Objectives Parts of a Radical Simplifying...

Radicals

07/27/12 lntaylor ©

Table of Contents

Learning Objectives

Parts of a Radical

Simplifying Radicals

Radical Expressions

Estimating Radicals

Practice

07/27/12 lntaylor ©

LO1:

LO2:

Define and differentiate radicals, radicands and root index

Estimate and simplify radical expressions

07/27/12 lntaylor ©TOC

Learning Objectives

PK1: Knowledge of exponent operations


Previous Knowledge

Def1:

Def2:

Radical is a symbol telling you to determine the root of a number

Radicand is the quantity under the radical ;


Parts of a Radical Expression(Definitions)

Def3: Root index is the number or letter to the left and above the radical (cube root of 8); (nth root of e)No index with a radical assumes square root ()


Simplifying Radicals

Step1: To simplify a radical you must first know your perfect squares

Step2: Only then will you understand the square roots

Step3: Memorize the following charts (perfect squares and square roots)

07/27/12lntaylor © TOC

Squares Equivalents

> 4

>

>

>

>

>

9

16

25

36

49

> 64

> 81

> 100

>

>

>

>

>

121

144

169

196

225

clear answers

07/27/12lntaylor © TOC

Square Roots Equivalents

> 2

>

>

>

>

>

3

4

5

6

7

> 8

> 9

> 10

>

>

>

>

>

11

12

13

14

15 clear answers



> 2

>

>

>

>

>

3

4

5

6

7

> 8

> 9

> 10

>

>

>

11

12

13

note

Do you see that the square root of a number squared is that number?

clear answers

Step1:

Step2:

This radicand (24) is not a perfect squareTherefore – find the perfect square!

Start by factoring the radicand into a perfect square times a numberWhat are the factors of 24?


What is the ?

Step3: Hint: Only one combination includes a perfect squareRewrite the problem with 2 radicals

Step4: Simplify (reduce) the radical containing the perfect squareLeave the other alone

Step5: This is your final answer

24

24 = 2(12)

3(8)

4(6) 4(6)

=

= 2

= 2

Now you try

What is the ?


Step1:

Step2:

This radicand (28) is not a perfect squareTherefore – find the perfect square!

Start by factoring the radicand into a perfect square times a numberWhat are the factors of 28?


What is the ?

Step3: Hint: Only one combination includes a perfect squareRewrite the problem with 2 radicals

Step4: Simplify (reduce) the radical containing the perfect squareLeave the other alone


28

28 = 2(14)

3(not a whole number)

4(7) 4(7)

=

= 2

= 2

Now you try

What is the ?


Step1:

Step2:

This radicand is not a perfect squareTherefore – find the perfect square!

Start by factoring the radicand into a perfect square times a numberWhat are the factors of the radicand?


What is the ?

Step3: Hint: make sure you factor out all perfect squares!Rewrite the problem with 3 radicals

Step4: Simplify (reduce) any radicals containing the perfect squaresLeave the other alone


72

72 = 9(8)

9(4)(2) 9(4)(2)

=

= 3*2

= 6

Now you try

What is the ?


Step1:

Step2:




What is the ?

Step3: Hint: make sure you factor out all perfect squares!Rewrite the problem with 3 radicals



675

675 = 25(27)

25(9)(3) 25(9)(3)

=

= 5*3

= 15

Now you try

What is the ?


Step1:

Step2:




What is the ?

Step3: Hint: make sure you factor out all perfect squares!Rewrite the problem with radicals



160

160 = 16(10) 16(10) =

= 4

= 4


Radical Expressions

Step1: To simplify radical expressions you must first understand exponents

Step2: Only then will you understand the radical expressions

Step3: Memorize the following charts (square root exponents)



2

xy

> 2x

>

>

>

>

>

2x

2

2

2y

5xy

> 8

> 2xy

> 2xy

>

>

2

2y

note

Do you see that the square root of an even exponent is half the exponent; the square root of an odd exponent puts half the exponent outside the radical and leaves an exponent of 1 under the radical?

clear answers


Radical Expressions

Note: If you understood the preceding chart then you are ready to go on!

What is ()?


Step1:

Step2:

To do these properly you must “unpack” each part of the expressionThere are 2 parts here!

Start by factoring each radicand


What is ()?

Step3: Hint: make sure you have factored out all perfect squares!Simplify the problem


√𝟒 𝒙𝟐 ( )

2x (3y)

() = 6xy

6xy

Now you try!

What is ()?


Step1:

Step2:


Start by factoring each radicand


What is ()?

Step3: Hint: make sure you have factored out all perfect squares!Simplify the problem


√𝟐𝟒 𝒙𝟐 ( )

√𝟒𝐱𝟐√𝟔 (9y)

() = 18xy

9y (2x)

Now you try!

What is ÷ 4x)?


Step1:

Step2:


Start by factoring the perfect squares out of each radicand


What is ÷ 4x)?

Step3: Hint: make sure you have factored out all perfect squares!Rewrite the problemCancel any terms


√𝟒𝟖𝒙𝟑 ÷ 4x ()

√𝟏𝟔𝐱𝟐√𝟑𝒙 ÷ 4x 4x 4x (5y )

5y

Now you try!

What is ÷ )?


Step1:

Step2:


Start by factoring the perfect squares out of each radicand


What is ÷ )?

Step3: Hint: make sure you have factored out all perfect squares!Rewrite the problemCancel any terms


√𝟑𝟎𝟎𝒙𝟑 ÷ )√𝟏𝟎𝟎𝐱𝟐√𝟑 𝒙 ÷

10x 5y ( )

xy


Estimating Radicals

Step1:

Step2:

Estimating a radical is not difficultFirst figure out which two perfect squares it lies between

Factor the perfect squares onto a number line


What is ?

Step3: Find the range between the perfect square radicands(64 – 49)This becomes the denominatorSubtract the middle radicand (60) from the lower radicand (49)This becomes the numeratorEstimate the decimal equivalent of the fraction

Step4: is approximately 7.75This is your estimated answerWhat is the actual ?Pretty Close!

√𝟔𝟎√𝟒𝟗 √𝟔𝟒

7 8

64 – 49 = 15

__15

60 – 49 = 11

11 ≈ 0.757.75 = 7.74596

Now you try!

What is ?


Step1:

Step2:




What is ?

Step3: Find the range between the perfect square radicands(100 – 86)This becomes the denominatorSubtract the middle radicand (86) from the lower radicand (81)This becomes the numeratorEstimate the decimal equivalent of the fraction


√𝟖𝟔√𝟖𝟏 √𝟏𝟎𝟎

9 10

100 – 81 = 19

__19

86 – 81 = 5

5 ≈ 0.259.25 = 9.2736

Now you try!

What is ?


Step1:

Step2:




What is ?

Step3: Find the range between the perfect square radicandsThis becomes the denominatorSubtract the middle radicand from the lower radicand This becomes the numeratorEstimate the decimal equivalent of the fraction


√𝟐𝟎𝟎√𝟏𝟗𝟔 √𝟐𝟐𝟓

14 15

225 – 196 = 29

__29

200 – 196 = 4

4 ≈ 0.1414.14 = 14.142


Practice


Problem Answer

Estimate

Estimate

Simplify

Simplify

Simplify

Simplify

Estimate

Estimate

Simplify -2

> ≈ 5.78 >

>

>

>

>

≈ 7.14

8

2

2

2

> ≈ 4.44

> 9.9 y

> - 4

clear answers

Radicals 07/27/12lntaylor ©. Table of Contents Learning Objectives Parts of a Radical Simplifying...

Documents

Transcript of Radicals 07/27/12lntaylor ©. Table of Contents Learning Objectives Parts of a Radical Simplifying...