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
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Learning Objectives
PK1: Knowledge of exponent operations
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Previous Knowledge
Def1:
Def2:
Radical is a symbol telling you to determine the root of a number
Radicand is the quantity under the radical ;
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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 ()
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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)
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Squares Equivalents
> 4
>
>
>
>
>
9
16
25
36
49
> 64
> 81
> 100
>
>
>
>
>
121
144
169
196
225
clear answers
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Square Roots Equivalents
> 2
>
>
>
>
>
3
4
5
6
7
> 8
> 9
> 10
>
>
>
>
>
11
12
13
14
15 clear answers
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Square Roots Equivalents
> 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?
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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 ?
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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?
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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
28
28 = 2(14)
3(not a whole number)
4(7) 4(7)
=
= 2
= 2
Now you try
What is the ?
07/26/12 lntaylor ©TOC
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?
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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
Step5: This is your final answer
72
72 = 9(8)
9(4)(2) 9(4)(2)
=
= 3*2
= 6
Now you try
What is the ?
07/26/12 lntaylor ©TOC
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?
07/27/12 lntaylor ©TOC
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
Step5: This is your final answer
675
675 = 25(27)
25(9)(3) 25(9)(3)
=
= 5*3
= 15
Now you try
What is the ?
07/26/12 lntaylor ©TOC
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?
07/27/12 lntaylor ©TOC
What is the ?
Step3: Hint: make sure you factor out all perfect squares!Rewrite the problem with radicals
Step4: Simplify (reduce) any radicals containing the perfect squaresLeave the other alone
Step5: This is your final answer
160
160 = 16(10) 16(10) =
= 4
= 4
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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)
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Square Roots Equivalents
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
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Radical Expressions
Note: If you understood the preceding chart then you are ready to go on!
What is ()?
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Step1:
Step2:
To do these properly you must “unpack” each part of the expressionThere are 2 parts here!
Start by factoring each radicand
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What is ()?
Step3: Hint: make sure you have factored out all perfect squares!Simplify the problem
Step4: This is your final answer
√𝟒 𝒙𝟐 ( )
2x (3y)
() = 6xy
6xy
Now you try!
What is ()?
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Step1:
Step2:
To do these properly you must “unpack” each part of the expressionThere are 2 parts here!
Start by factoring each radicand
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What is ()?
Step3: Hint: make sure you have factored out all perfect squares!Simplify the problem
Step4: This is your final answer
√𝟐𝟒 𝒙𝟐 ( )
√𝟒𝐱𝟐√𝟔 (9y)
() = 18xy
9y (2x)
Now you try!
What is ÷ 4x)?
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Step1:
Step2:
To do these properly you must “unpack” each part of the expressionThere are 2 parts here!
Start by factoring the perfect squares out of each radicand
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What is ÷ 4x)?
Step3: Hint: make sure you have factored out all perfect squares!Rewrite the problemCancel any terms
Step4: This is your final answer
√𝟒𝟖𝒙𝟑 ÷ 4x ()
√𝟏𝟔𝐱𝟐√𝟑𝒙 ÷ 4x 4x 4x (5y )
5y
Now you try!
What is ÷ )?
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Step1:
Step2:
To do these properly you must “unpack” each part of the expressionThere are 2 parts here!
Start by factoring the perfect squares out of each radicand
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What is ÷ )?
Step3: Hint: make sure you have factored out all perfect squares!Rewrite the problemCancel any terms
Step4: This is your final answer
√𝟑𝟎𝟎𝒙𝟑 ÷ )√𝟏𝟎𝟎𝐱𝟐√𝟑 𝒙 ÷
10x 5y ( )
xy
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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
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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 ?
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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
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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
Step4: is approximately 9.25This is your estimated answerWhat is the actual ?Pretty Close!
√𝟖𝟔√𝟖𝟏 √𝟏𝟎𝟎
9 10
100 – 81 = 19
__19
86 – 81 = 5
5 ≈ 0.259.25 = 9.2736
Now you try!
What is ?
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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
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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
Step4: is approximately 14.14This is your estimated answerWhat is the actual ?Pretty Close!
√𝟐𝟎𝟎√𝟏𝟗𝟔 √𝟐𝟐𝟓
14 15
225 – 196 = 29
__29
200 – 196 = 4
4 ≈ 0.1414.14 = 14.142
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Practice
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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
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