CHAPTER 12: RADICALS Contents - Santiago Canyon CollegeΒ Β· 2017-08-21Β Β· D. SIMPLIFY RADICALS WITH...
Transcript of CHAPTER 12: RADICALS Contents - Santiago Canyon CollegeΒ Β· 2017-08-21Β Β· D. SIMPLIFY RADICALS WITH...
Chapter 12
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CHAPTER 12: RADICALS Chapter Objectives
By the end of this chapter, students should be able to: Simplify radical expressions Rationalize denominators (monomial and binomial) of radical expressions Add, subtract, and multiply radical expressions with and without variables Solve equations containing radicals
Contents CHAPTER 12: RADICALS ............................................................................................................................ 317
SECTION 12.1 INTRODUCTION TO RADICALS ...................................................................................... 319
A. INTRODUCTION TO PERFECT SQUARES AND PRINCIPAL SQUARE ROOT ............................... 319
B. INTRODUCTION TO RADICALS ................................................................................................. 320
C. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ππππππ ROOT .................................................... 322
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ππππππ ROOT USING EXPONENT RULE ............ 323
E. SIMPLIFY RADICALS WITH NO PERFECT ROOT ........................................................................ 325
F. SIMPLIFY RADICALS WITH COEFFICIENTS ................................................................................ 326
G. SIMPLIFY RADICALS WITH VARIABLES WITH NO PERFECT RADICANTS ................................. 327
EXERCISE ........................................................................................................................................... 328
SECTION 12.2: ADD AND SUBTRACT RADICALS ................................................................................... 329
A. ADD AND SUBTRACT LIKE RADICALS ....................................................................................... 329
B. SIMPLIFY, THEN ADD AND SUBTRACT LIKE RADICALS ............................................................ 330
EXERCISE ........................................................................................................................................... 331
SECTION 12.3: MULTIPLY AND DIVIDE RADICALS ............................................................................... 332
A. MULTIPLY RADICALS WITH MONOMIALS ................................................................................ 332
B. DISTRIBUTE WITH RADICALS .................................................................................................... 334
C. MULTIPLY RADICALS USING FOIL ............................................................................................. 335
D. MULTIPLY RADICALS WITH SPECIAL-PRODUCT FORMULAS ................................................... 336
E. SIMPLIFY QUOTIENTS WITH RADICALS .................................................................................... 337
EXERCISE ........................................................................................................................................... 339
SECTION 12.4: RATIONALIZE DENOMINATORS ................................................................................... 341
A. RATIONALIZING DENOMINATORS WITH SQUARE ROOTS ...................................................... 341
B. RATIONALIZING DENOMINATORS WITH HIGHER ROOTS ....................................................... 342
C. RATIONALIZE DENOMINATORS USING THE CONJUGATE ....................................................... 343
EXERCISE ........................................................................................................................................... 345
SECTION 12.5: RADICAL EQUATIONS ................................................................................................... 346
Chapter 12
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A. RADICAL EQUATIONS WITH SQUARE ROOTS .......................................................................... 346
B. RADICAL EQUATIONS WITH TWO SQUARE ROOTS ................................................................. 348
C. RADICAL EQUATIONS WITH HIGHER ROOTS ........................................................................... 351
EXERCISE ........................................................................................................................................... 352
CHAPTER REVIEW ................................................................................................................................. 353
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SECTION 12.1 INTRODUCTION TO RADICALS A. INTRODUCTION TO PERFECT SQUARES AND PRINCIPAL SQUARE ROOT
MEDIA LESSON Introduction to square roots (Duration 7:03 )
View the video lesson, take notes and complete the problems below
Some numbers are called _________________________________. It is important that we can recognize
________________________________ when working with square roots.
12 = 1 β 1 = ___________________ 62 = 6 β 6 =___________________
22 = 2 β 2 = ___________________ 72 = 7 β 7 = ___________________
32 = 3 β 3 = ___________________ 82 = 8 β 8 =___________________
42 = 4 β 4 = ___________________ 92 = 9 β 9 =___________________
52 = 5 β 5 = ___________________ 102 = 10 β 10 =___________________
To determine the square root of a number, we have a special symbol.
β9
The square root of a number is the number times itself that equals the given number.
β9 = ____________________________________________________________
β36 = ____________________________________________________________
β49 = ____________________________________________________________
β81 =____________________________________________________________
You can think of the square root as the opposite or inverse of squaring.
Actually, numbers have two square roots. One is positive and one is negative.
5 β 5 = 25 and β5 β β5 = 25
To avoid confusion
β25 = 5 and ββ25 = β5
What about these square roots?
β20
β61
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YOU TRY
a) Find the perfect square of:
112 = ________________ 122 = ________________ 132 = ________________ 142 = ________________ 152 = ________________ 162 = ________________ 172 = ________________ 182 = ________________ 192 =________________ 202 =________________
b) Find the square root of:
β441 = _______________ β484 =_______________
β529 =_______________
β576 =_______________ β625 =_______________ β676 =_______________ β729 =_______________
β784 =_______________ β841 = _______________
β900 = _______________
MEDIA LESSON Principal nth square roots vs. general square roots (Duration 5:23 )
Note: In this class, we will only consider the principal ππππππroots when we discuss radicals.
B. INTRODUCTION TO RADICALS Radicals are a common concept in algebra. In fact, we think of radicals as reversing the operation of an exponent. Hence, instead of the βsquareβ of a number, we βsquare rootβ a number; instead of the βcubeβ of a number, we βcube rootβ a number to reverse the square to find the base. Square roots are the most common type of radical used in algebra.
Definition
If ππ is a positive real number, then the principal square root of a number ππ is defined as
βππ = ππ if and only if ππ = ππππ
The β is the radical symbol, and ππ is called the radicand.
If given something like βππππ, then 3 is called the root or index; hence, βππ ππ
is called the cube root or third root of ππ. In general,
βππππ = ππ if and only if ππ = ππππ
If ππ is even, then ππ and ππ must be greater than or equal to zero. If ππ is odd, then ππ and ππ must be any real number.
Here are some examples of principal square roots:
β1 = 1 β121 = 11 β4 = 2 β625 = 25 β9 = 3 ββ81 is not a real number
The final example ββ81 is not a real number. Since square root has the index is 2, which is even, the radicand must be greater than or equal to zero and since β81 < 0, then there is no real number in which we can square and will result in β81,i.e., ?2 = β81. So, for now, when we obtain a radicand that is negative and the root is even, we say that this number is not a real number. There is a type of number where we can evaluate these numbers, but just not a real one.
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MEDIA LESSON Introduction to square roots, cube roots, and Nth roots (Duration 9:09)
View the video lesson, take notes and complete the problems below
The principal ππππππ root of ππ is the ππππππ root that has the same sign as ππ, and it is denoted by the radical symbol.
βππππ We read this as the β___________________________β, β______________β, or β_______________β. The positive integer ______________________________ of the radical. If ππ = 2, ____________ the index.
The number _______________________.
β4 =________________
β164 =_______________
ββ4 =________________
ββ164 =_______________
Square roots (n = 2) β1 =________________________________ ββ1 =________________________________
β4 = ________________________________ ββ4 = ________________________________
β9 = ________________________________ ββ9 = ________________________________
β16 = _______________________________ ββ16 = _______________________________
β25 = _______________________________ ββ25 = _______________________________
Cube roots (n = 3)
β13 = __________________________ ββ13 = __________________________
β83 = __________________________ ββ83 = __________________________
β273 =__________________________ ββ273 =_________________________
β643 = __________________________ ββ643 = _________________________
β1253 =_________________________ ββ1253 =________________________
Example: Simplify
1) β36 =
2) ββ81 =
3) οΏ½49 =
4) β643 =
5) β325 = 6) β ββ83 =
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Inverse properties of ππππππ Powers and ππππππ Roots
If ππ has a principal ππππππ root, then____________________________.
If ππ is odd, then ______________________________. If ππ is even, then ______________________________. We need the ____________________________ for any ππππππ root with an _____________ exponent
for which the index is ____________ to assure the ππππππ root is ______________.
Example: Simplify
1) βπ₯π₯2
2) βπ₯π₯93
3) βπ₯π₯84
4) οΏ½π¦π¦124
C. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ππππππ ROOT
MEDIA LESSON Simplify perfect ππππππroots (Duration 4:04 )
View the video lesson, take notes and complete the problems below
Example: a) β81
b) β273
c) β164
d) β243
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MEDIA LESSON Simplify perfect ππππππroots β negative radicands (Duration 4:32 )
View the video lesson, take notes and complete the problems below
Example: Simplify each of the following.
a) β164 = ________________________________________________________________________
b) ββ325 = ________________________________________________________________________
c) ββ646 = ________________________________________________________________________
YOU TRY Simplify. Show your work.
a) ββ36
b) ββ64 3
c) β β6254
d) β15
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ππππππ ROOT USING EXPONENT RULE
There is a more efficient way to find the πππ‘π‘β root by using the exponent rule but first letβs learn a different method of prime factorization to factor a large number to help us break down a large number into primes. This alternative method to a factor tree is called the βstacked divisionβ method.
MEDIA LESSON Prime factorization β stacked division method (Duration 3:45)
View the video lesson, take notes and complete the problems below
a) 1,350 b) 168
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MEDIA LESSON Simplify perfect root radicals using the exponent rule (Duration 5:00 )
View the video lesson, take notes and complete the problems below
Roots: βππππ where ππ is the _______________
Roots of an expression with exponents: _________________the ________________ by the __________.
Example: Simplify.
a) οΏ½46,656 = b) οΏ½1,889,5685 =
MEDIA LESSON Simplify perfect root radicals with variables (Duration 5:43 )
View the video lesson, take notes and complete the problems below
Example: Simplify.
a) βπ§π§93
b) βππ6
c) ββππ105
YOU TRY Simplify the following radicals using the exponent rule. Show your work.
a) β646
b) β7293
c) οΏ½π₯π₯2π¦π¦4π§π§10
d) οΏ½π₯π₯21π¦π¦427
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E. SIMPLIFY RADICALS WITH NO PERFECT ROOT Not all radicands are perfect squares, where when we take the square root, we obtain a positive integer. For example, if we input β8 in a calculator, the calculator would display
2.828427124746190097603377448419β¦ and even this number is a rounded approximation of the square root. To be as accurate as possible, we will leave all answers in exact form, i.e., answers contain integers and radicals β no decimals. When we say to simplify an expression with radicals, the simplified expression should have
β’ a radical, unless the radical reduces to an integer β’ a radicand with no factors containing perfect squares β’ no decimals
Following these guidelines ensures the expression is in its simplest form.
Product rule for radicals
If ππ,ππ are any two positive real numbers, then
βππππ = βππ β βππ In general, if ππ,ππ are any two positive real numbers, then
βππππππ = βππππ β βππππ
Where ππ is a positive integer and ππ β₯ ππ.
MEDIA LESSON Simplify square roots with not perfect square radicants (Duration 7:03)
View the video lesson, take notes and complete the problems below
Recall: The square root of a square
For a non-negative real number, ππ: βππππ = ππ
For example: β25 = β5 β 5 = β52 = 5 The product rule for square roots
Given that ππ and ππ are non-negative real numbers, ___________________________________________.
β45 = ________________________________________________________________________. Example: β8 = _____________________________________________________________
β48 = _____________________________________________________________
β150 = _____________________________________________________________
οΏ½1,350 = _____________________________________________________________
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MEDIA LESSON Simplify radicals with not perfect radicants β using exponent rule (Duration 4:22)
View the video lesson, take notes and complete the problems below
To take roots we _______________ the ______________ by the index
βππ2ππ =
βππππππππ = When we divide if there is a remainder, the remainder ________________________________________.
Example:
a) β72 b) β7503
YOU TRY
Simplify. Show your work.
a) β75
b) β2003
F. SIMPLIFY RADICALS WITH COEFFICIENTS
MEDIA LESSON Simplify radicals with coefficients (Duration 3:52)
View the video lesson, take notes and complete the problems below
If there is a coefficient on the radical: ______________________ by what ________________________. Example: a) β8β600 b) 3 ββ965
YOU TRY
Simplify. a) 5β63
b) β8β392
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G. SIMPLIFY RADICALS WITH VARIABLES WITH NO PERFECT RADICANTS
MEDIA LESSON Simplify radicals with variables (Duration 4:22)
View the video lesson, take notes and complete the problems below
Variable in radicals: _____________________ the __________________ by the ___________________
Remainders: ________________________________________________
Example:
a) βππ13ππ23ππ10ππ34
b) οΏ½125π₯π₯4π¦π¦π§π§5
YOU TRY
Simplify. Assume all variables are positive. a) οΏ½π₯π₯6π¦π¦5
b) β5οΏ½18π₯π₯4π¦π¦6π§π§10
c) οΏ½20π₯π₯5π¦π¦9π§π§6
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EXERCISE Simplify. Show all your work. Assume all variables are positive.
1) β245
2) β36
3) β12
4) 3β12
5) 6β128
6) β8β392
7) β192ππ
8) β196π£π£2
9) β252π₯π₯2
10) ββ100ππ4
11) β7β64π₯π₯4
12) β5β36ππ
13) β4οΏ½175ππ4
14) 8οΏ½112ππ2
15) β2β128ππ
16) οΏ½45π₯π₯2π¦π¦2
17) οΏ½16π₯π₯3π¦π¦3
18) οΏ½320π₯π₯4π¦π¦4
19) βοΏ½32π₯π₯π¦π¦2π§π§3
20) 5οΏ½245π₯π₯2π¦π¦3
21) β2β180π’π’3π£π£
22) β72ππ3ππ4
23) 2οΏ½80βππ4ππ
24) 6β50ππ4ππππ2
25) 8β98ππππ
26) β512ππ4ππ2
27) β100ππ4ππ3
28) β8οΏ½180π₯π₯4π¦π¦2π§π§4
29) 2οΏ½72π₯π₯2π¦π¦2
30) β5οΏ½36π₯π₯3π¦π¦4
Simplify. Show all your work. Assume all variables are positive.
31) β6253
32) β7503
33) β8753
34) β4 β964
35) 6 β1124
36) β648ππ24
37) β224ππ35
38) οΏ½224ππ55
39) β3 β896ππ7
40) β2 ββ48π£π£73
41) β7 β320ππ63
42) οΏ½β135π₯π₯5π¦π¦33
43) οΏ½β32π₯π₯4π¦π¦43
44) οΏ½256π₯π₯4π¦π¦63
45) 7 οΏ½β81π₯π₯3π¦π¦73
46) 2 β375π’π’2π£π£83
47) β3 β192ππππ23
48) 6 οΏ½β54ππ8ππ3ππ73
49) 6 οΏ½648π₯π₯5π¦π¦7π§π§24 50) 9οΏ½9π₯π₯2π¦π¦5π§π§3
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SECTION 12.2: ADD AND SUBTRACT RADICALS Adding and subtracting radicals are very similar to adding and subtracting with variables. In order to combine terms, they need to be like terms. With radicals, we have something similar called like radicals. Letβs look at an example with like terms and like radicals.
2π₯π₯ + 5π₯π₯ (2 + 5)π₯π₯
7π₯π₯
2β3 + 5β3 (2 + 5)β3
7β3 Notice that when we combined the terms with β3, it was similar to combining terms with π₯π₯. When adding and subtracting with radicals, we can combine like radicals just as like terms.
Definition
If two radicals have the same radicand and the same root, then they are called like radicals. If this is so, then
ππβππ Β± ππβππ = (ππ Β± ππ)βππ,
Where ππ,ππ are real numbers and ππ is some positive real number.
In general, for any root ππ, ππβππππ Β± ππβππππ = (ππ Β± ππ)βππππ ,
Where ππ,ππ are real numbers and ππ is some positive real number.
Note: When simplifying radicals with addition and subtraction, we will simplify the expression first, and then reduce out any factors from the radicand following the guidelines in the previous section.
A. ADD AND SUBTRACT LIKE RADICALS
MEDIA LESSON Add and subtract like radicals (Duration 3:11)
View the video lesson, take notes and complete the problems below
Simplify: 2π₯π₯ β 5π¦π¦ + 3π₯π₯ + 2π¦π¦
_______________________
Simplify: 2β3 β 5β7 + 3β3 + 2β7
_______________________
When adding and subtracting radicals, we can ______________________________________________. Example:
a) β4β6 + 2β11 + β11 β 5β6 b) β53 + 3β5 β 8β53 + 2β5
YOU TRY
Simplify
a) 7β65 + 4β35 β 9β35 + β65
b) β3β2 + 3β5 + 3β5
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B. SIMPLIFY, THEN ADD AND SUBTRACT LIKE RADICALS
MEDIA LESSON Add or subtract radicals requiring simplifying first (Duration 3:46)
View the video lesson, take notes and complete the problems below
Guidelines for adding and subtracting radicals
1. ______________________________________________________________________________
2. ______________________________________________________________________________
3. ______________________________________________________________________________
Example: Simplify β2οΏ½50π₯π₯5 + 5οΏ½18π₯π₯5 50
/\ 18 /\
MEDIA LESSON Add or subtract radicals requiring simplifying first (continue) (Duration 5:12)
View the video lesson, take notes and complete the problems below
Example: a) 2β18 + β50 b) π₯π₯ οΏ½π₯π₯2π¦π¦53 + π¦π¦ οΏ½π₯π₯5π¦π¦23
YOU TRY
Simplify.
a) 5β45 + 6β18 β 2β98 + β20
b) 4β543 β 9β163 + 5β93
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EXERCISE Simplify. In this section, we assume all variables to be positive.
1) 2β5 + 2β5 + 2β5
2) β2β6 β 2β6 ββ6
3) 3β6 + 3β5 + 2β5
4) 2β2 β 3β18 β β2
5) 3β2 + 2β8 β 3β18
6) β3β6 ββ12 + 3β3
7) 3β18 β β2 β 3β2
8) β2β18β 3β8 β β20 + 2β20
9) β2β24β 2β6 + 2β6 + 2β20
10) 3β24 β 3β27 + 2β6 + 2β8
11) β2β163 + 2β163 + 2β23
12) 2β2434 β 2β2434 β β34
13) β6254 -5β6254 + β643 β 5β643
14) 3β24 β 2β24 β β2434
15) ββ3244 + 3β3244 β 3β44
16) 2β24 + 2β34 + 3β644 β β34
17) β3β65 β β645 + 2β1925 β 2β645
18) 2β1605 β 2β1925 β β1605 β ββ1605
19) ββ2566 β 2β46 β 3β3206 β 2β1286
20) 3β1353 β β813 β β1353
21) β3β18π₯π₯5 β β8π₯π₯5 + 2β8π₯π₯5 + 2β8π₯π₯5
22) β2οΏ½2π₯π₯π¦π¦ β οΏ½2π₯π₯π¦π¦ + 3οΏ½8π₯π₯π¦π¦ + 3οΏ½8π₯π₯π¦π¦
23) 2β6π₯π₯2 β β54π₯π₯2 β 3οΏ½27π₯π₯2π¦π¦ β οΏ½3π₯π₯2π¦π¦
24) 2π₯π₯οΏ½20π¦π¦2 + 7π¦π¦β20π₯π₯2 β οΏ½3π₯π₯π¦π¦
25) 3β24π‘π‘ β 3β54π‘π‘ β 2β96π‘π‘ + 2β150π‘π‘
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SECTION 12.3: MULTIPLY AND DIVIDE RADICALS
Recall the product rule for radicals in the previous section:
Product rule for radicals
If ππ,ππ are any two positive real numbers, then
βππππ = βππ β βππ In general, if ππ,ππ are any two positive real numbers, then
βππππππ = βππππ β βππππ
Where ππ is a positive integer and ππ β₯ ππ.
As long as the roots of each radical in the product are the same, we can apply the product rule and then simplify as usual. At first, we will bring the radicals together under one radical, then simplify the radical by applying the product rule again.
A. MULTIPLY RADICALS WITH MONOMIALS
MEDIA LESSON Multiply monomial radical expressions (Duration 10:32 )
View the video lesson, take notes and complete the problems below
To multiply two radicals with the same index. Multiply the _________________________together and
multiply the ____________________ together. Then simplify.
Product rule (with coefficients): pβπ’π’ππ β ππ βπ£π£ππ = ________________
Example 1: β2 β β3 = ______________________________________
Example 2: 3β53 β 4β73 = ____________________________________
Multiply:
a) β15 β β6 b) β183 β β603
c) 3β12 β 5β63
d) β2β404 β 7β184
e) ββ6 Β· β3β6
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YOU TRY
Simplify: a) β5β14 β 4β6
b) 2 β183 β 6 β153
Note: In this section, we assume all variables to be positive.
MEDIA LESSON Multiply monomial radicals with variables (Duration 4:58 )
View the video lesson, take notes and complete the problems below
Example: Multiply.
a) β18π₯π₯3 β β30π₯π₯2 b) β16π₯π₯23 β β81π₯π₯23
YOU TRY
Simplify.
a) β8π₯π₯25 β β4π₯π₯35
b) β60π₯π₯4 β β6π₯π₯7
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B. DISTRIBUTE WITH RADICALS When there is a term in front of the parenthesis, we distribute that term to each term inside the parenthesis. This method is applied to radicals.
MEDIA LESSON Multiply square roots using Distributive property (Duration 2:25 )
View the video lesson, take notes and complete the problems below
Example: β7οΏ½β14 β β2οΏ½ β3οΏ½5 + β3οΏ½
MEDIA LESSON Multiplying radical expressions with variables using Distributive property (Duration 6:57 )
View the video lesson, take notes and complete the problems below
Example: a) βπ₯π₯οΏ½2βπ₯π₯ β 3οΏ½
b) 4οΏ½π¦π¦οΏ½5οΏ½π₯π₯π¦π¦3 β οΏ½π¦π¦3οΏ½
c) βπ§π§3 οΏ½βπ§π§23 β 7βπ§π§53 + 2 βπ§π§83 οΏ½
YOU TRY
Simplify.
a) 7β6 (3β10 β 5β15)
b) β3οΏ½7β15π₯π₯3 + 8π₯π₯β60π₯π₯οΏ½
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C. MULTIPLY RADICALS USING FOIL
MEDIA LESSON Multiply binomials with radicals (Duration 4:10)
View the video lesson, take notes and complete the problems below
Recall: (ππ + ππ)(ππ + ππ) = ____________________________________
Always be sure your final answer is ____________________________.
Example: a) οΏ½3β7 β 2β5οΏ½οΏ½β7 + 6β5οΏ½
b) οΏ½2 β93 + 5οΏ½ οΏ½4 β33 β 1οΏ½
MEDIA LESSON Multiply binomials with radicals with variables (Duration 5:29)
View the video lesson, take notes and complete the problems below
Example: a) οΏ½2βπ₯π₯ + 3οΏ½οΏ½5βπ₯π₯ β 4οΏ½ b) οΏ½3π₯π₯2 + βπ₯π₯23 οΏ½ οΏ½2 βπ₯π₯3 β 1οΏ½
YOU TRY
Simplify.
a) (β5 β 2β3)(4β10 + 6β6)
b) οΏ½3βπ£π£ + 2β3οΏ½οΏ½5βπ£π£ β 7β3οΏ½
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D. MULTIPLY RADICALS WITH SPECIAL-PRODUCT FORMULAS
MEDIA LESSON Multiply radicals using the perfect square formula (Duration 3:44)
View the video lesson, take notes and complete the problems below
Recall the Perfect Square formula: (ππ + ππ)2 = ________________________________
Always be sure your final answer is _________________________
Example:
a) οΏ½β6 ββ2οΏ½2
b) οΏ½2 + 3β7οΏ½2
Conjugates
Recall the Difference of for two squares formula: (ππ β ππ)(ππ + ππ) = ππππ β ππππ Notice in the 2 factors (ππ β ππ) and (ππ + ππ) have the same first and second term but there is a sign change in the middle. When we have 2 binomials like that, we say they are conjugates of each other. Example:
Binomials Its conjugate 3 β 5 3 + 5 π₯π₯ + 5 π₯π₯ β 5
1 β β2 1 + β2 The product of two conjugates is the Difference of two squares. This result is very helpful when multiplying radical expressions and rationalizing radicals in the later section of this chapter.
MEDIA LESSON Multiply radicals using the difference of squares formula (Duration 1:27)
View the video lesson, take notes and complete the problems below
The Difference of Squares formula: (ππ β ππ)(ππ + ππ) = ____________________________________
οΏ½3 β β6οΏ½οΏ½3 + β6οΏ½ = ____________________________________________________________________
οΏ½β2 ββ5οΏ½οΏ½β2 + β5οΏ½ = _________________________________________________________________
οΏ½2β3 + 3β7οΏ½οΏ½2β3 β 3β7οΏ½ = ____________________________________________________________
= ____________________________________________________________
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YOU TRY
a) Simplify: (5β7 + β2)2
b) Simplify: (8 ββ5)(8 + β5)
E. SIMPLIFY QUOTIENTS WITH RADICALS
Quotient rule for radicals
If ππ,ππ are any two positive real numbers, where ππ β ππ, then
οΏ½ππππ
=βππβππ
If ππ,ππ are any two positive real numbers, where ππ β ππ, then
οΏ½ππππ
ππ=βππππ
βππππ
Where ππ is a positive integer and ππ β₯ ππ.
MEDIA LESSON Divide radicals (Duration 3:44)
View the video lesson, take notes and complete the problems below
Note: A rational expression is not considered simplified if there is a fraction under the radical or if there is a radical in the denominator.
Example:
a) οΏ½7516
b) οΏ½3244
3
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MEDIA LESSON Divide radicals with variables (Duration 4:34 )
View the video lesson, take notes and complete the problems below
Examples:
a) οΏ½100π₯π₯5π₯π₯
, assume π₯π₯ is positive
b) οΏ½64π₯π₯2π¦π¦53
οΏ½4π¦π¦23 , assume π¦π¦ is not 0
MEDIA LESSON Divide expressions with radicals (Duration 4:20 )
View the video lesson, take notes and complete the problems below
Simplify expressions with radicals: Always _______________________the _____________________ first Before ____________________ with fractions, be sure to __________________ first! Examples:
a) 15 + β175
10
b) 8 β β48
6
YOU TRY
Simplify.
a) β3+β27
3
b) οΏ½44π¦π¦6ππ4
οΏ½9π¦π¦2ππ8
c) 15 β1083
20 β23
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EXERCISE Simplify. Assume all variables are positive.
1) β4β16 β 3β5
2) 3β10 β β20
3) β5β10ππ2 β β5ππ3
4) β12ππ β β15ππ
5) 3β4ππ43 β β10ππ33
6) β4π₯π₯33 β β2π₯π₯43
7) β6(β2 + 2)
8) 5β10(5ππ + β2)
9) β5β15(3β3 + 2)
10) 5β15(3β3 + 2)
11) β10(β5 + β2)
12) β15(β5 β 3β3π£π£)
13) (2 + 2οΏ½2)(β3 + β2)
14) (β2 + β3)(β5 + 2β3)
15) (β5 β 4β3)(β3β 4β3)
16) (β5 β 5)(2β5 β 1)
17) (β2ππ + 2β3ππ)(3β2ππ + β5ππ)
18) (5β2 β 1)(ββ2ππ + 5)
19) β10β6
20) β5
4β125
21) β125β100
22) β53
4 β43
23) 2β43β3
24) 3 β103
5 β273
25) οΏ½12ππ2
οΏ½3ππ
26) 4+ 8β452β4
27) 3+ β12β3
28) 4β2β23β32
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29) 4ββ30β15
30) 5 β5ππ44
β8ππ24
31) 5π₯π₯2
4π₯π₯π¦π¦οΏ½3π₯π₯π¦π¦ 32) (5 + 2β6)2
33) (π₯π₯ β π₯π₯β5)2 34) (β3 β β7)2
35) (5β6 + 2β3)2
36) (β2 β β5)(β2 + β5)
37) (βπ₯π₯ β οΏ½π¦π¦)(βπ₯π₯ + οΏ½π¦π¦)
38) (4 β 2β3)(4 + 2β3)
39) (π₯π₯ β π¦π¦β3)(π₯π₯ + π¦π¦β3)
40) (9βπ₯π₯ + οΏ½π¦π¦)(9βπ₯π₯ β οΏ½π¦π¦)
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SECTION 12.4: RATIONALIZE DENOMINATORS A. RATIONALIZING DENOMINATORS WITH SQUARE ROOTS
Rationalizing the denominator with square roots
To rationalize the denominator with a square root, multiply the numerator and denominator by the exact radical in the denominator, e.g.,
ππβππ
ββππβππ
MEDIA LESSON Rationalize monomials (Duration 3:42)
View the video lesson, take notes and complete the problems below
Example: Simplify by rationalizing the denominator. a) 20
β10 b) 35
3β7
MEDIA LESSON Rationalize monomials with variables (Duration 4:58)
View the video lesson, take notes and complete the problems below
Rationalize denominators: No _________________________ in the _____________________________
To clear radicals: ___________by the extra needed factors in denominator (multiply by the same on top!)
It may be helpful to __________________ first (both _________________ and ___________________).
Example:
a) β7ππππβ6ππππ2
b) οΏ½ 5π₯π₯π¦π¦3
15π₯π₯π¦π¦π₯π₯
YOU TRY
Simplify.
a) β6β5
b) 6β1412β22
c) β3β92β6
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B. RATIONALIZING DENOMINATORS WITH HIGHER ROOTS Radicals with higher roots in the denominators are a bit more challenging. Notice, rationalizing the denominator with square roots works out nicely because we are only trying to obtain a radicand that is a perfect square in the denominator. When we rationalize higher roots, we need to pay attention to the index to make sure that we multiply enough factors to clear them out of the radical.
MEDIA LESSON Rationalize higher roots (Duration 4:20)
View the video lesson, take notes and complete the problems below
Rationalize β Monomial higher root
Use the ____________________
To clear radicals _____________ by extra needed factors in denominator (multiply by the same on top!)
Hint: ___________________ numbers!
Example:
a) 5βππ27
b) οΏ½ 79ππ2ππ
3
YOU TRY
Simplify.
a) 4 β23
7 β253
b) 3 β114
β24
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C. RATIONALIZE DENOMINATORS USING THE CONJUGATE There are times where the given denominator is not just one term. Often, in the denominator, we have a difference or sum of two terms in which one or both terms are square roots. In order to rationalize these denominators, we use the idea from a difference of two squares:
(ππ + ππ)(ππ β ππ) = ππ2 β ππ2
Rationalize denominators using the conjugate
We rationalize denominators of the type ππ Β± βππ by multiplying the numerator and denominator by their conjugates, e.g.,
1ππ + βππ
βππ β βππππ β βππ
=ππ β βππ
(ππ)2 β (βππ)2
The conjugate for β’ ππ + βππ is ππ β βππ β’ ππ β βππ is ππ + βππ
The case is similar for when there is something like βππ Β± βππ in the denominator.
MEDIA LESSON Rationalize denominators using the conjugate (Duration 4:56)
View the video lesson, take notes and complete the problems below
Rationalize β Binomials
What doesnβt work: 1
2+β3
Recall: οΏ½2 + β3οΏ½ _______________________
Multiply by the ________________________
Example:
a) 6
5ββ3 b)
3β5β24+2β2
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MEDIA LESSON Rationalize denominators using the conjugate (Duration 2:59)
View the video lesson, take notes and complete the problems below
Example: Rationalize the denominator.
a) β2
4+β10
YOU TRY
Simplify.
a) 2
β3β5
b) 3ββ52ββ3
c) 2β5β3β75β6+4β2
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EXERCISE Simplify. Assume all variables are positive.
1) 2β43β3
2) β12β3
3) β23β5
4) 4β3β15
5) 4+2β3β9
6) β53
4 β43
7) 2β23 8)
6 β23
β93 9) 8
β3π₯π₯23
10) 2π₯π₯βπ₯π₯3 11)
π£π£β2π£π£34 12)
1β5π₯π₯4
13) 4+2β35β4
14) 2β5β54β13
15) β2β3β3
β3
16) 5
3β5+β2 17)
25+β2
18) 3
4β3β3
19) 4
3+β5 20) β 4
4β4β2 21)
45 + β5π₯π₯2
22) 5
2+β5ππ3 23)
2ββ5β3+β5
24) β3+β22β3ββ2
25) 4β2+33β2+β3
26) 5
β3+4β5 27)
2β5+β31ββ3
28) ππβππ
βππββππ 29)
7βππ+βππ
30) ππββππππ+βππ
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SECTION 12.5: RADICAL EQUATIONS Here we look at equations with radicals. As you might expect, to clear a radical we can raise both sides to an exponent. Recall, the roots of radicals can be thought of reversing an exponent. Hence, to reverse a radical, we will use exponents.
Solving radical equations
If ππ β₯ ππ and ππ β₯ ππ, then
βππ = ππ if and only if ππ = ππππ If ππ β₯ ππ and ππ is a real number, then
βππππ = ππ if and only if ππ = ππππ We assume in this chapter that all variables are greater than or equal to zero.
We can apply the following method to solve equations with radicals.
Steps for solving radical equations
Step 1. Isolate the radical.
Step 2. Raise both sides of the equation to the power of the root (index).
Step 3. Solve the equation as usual.
Step 4. Verify the solution(s). (Recall, we will omit any extraneous solutions.)
A. RADICAL EQUATIONS WITH SQUARE ROOTS
MEDIA LESSON Solve equations with one radical (Duration 6:47)
View the video lesson, take notes and complete the problems below
Solving equations having one radical
1. _________________ the radical on ____________________________________ of the equation.
2. ______________________________ of the equation to the _____________ of the __________.
3. ____________ the resulting equation.
4. __________________________________________. Some solutions might ________________.
The solutions that ________________________ are called ______________________ solutions.
οΏ½βπ₯π₯οΏ½2
= ___________ οΏ½βπ₯π₯3 οΏ½3
= ____________
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Example: Solve. a) βπ₯π₯ β 7 = 11
b) β3π₯π₯ + 2 β 7 = 0
c) 2β5π₯π₯ β 13 β 8 = 0
d) βπ₯π₯ + 6 = π₯π₯
YOU TRY
Solve for π₯π₯.
a) β7π₯π₯ + 2 = 4
b) βπ₯π₯ + 3 = 5
c) π₯π₯ + β4π₯π₯ + 1 = 5
d) βπ₯π₯ + 6 = π₯π₯ + 4
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B. RADICAL EQUATIONS WITH TWO SQUARE ROOTS
MEDIA LESSON Solve equations with two radicals (Duration 5:11)
View the video lesson, take notes and complete the problems below
Solving equations having two radicals
1. Put ______________________ on _____________________ of the ________________________.
2. __________________________________ to the ________________ of the _________________.
3. If one radical _______________, _____________ the remaining radical and raise ____________
_________________ to the ___________ of the index again. (If the radicals have been eliminated
skip this step.)
4. ______________ the resulting equation.
5. Check for ______________________________________.
Example: Solve.
a) β2π₯π₯ + 3 β βπ₯π₯ β 8 = 0 b) 3 + βπ₯π₯ β 6 = βπ₯π₯ + 9
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MEDIA LESSON Solve equations with two radicals β part 2 (Duration 4:33 )
View the video lesson, take notes and complete the problems below
Example: Solve the equation. β1 β 8π₯π₯ β ββ16π₯π₯ β 12 = 1
MEDIA LESSON Solve equations with two radicals β part 3 β check solutions (Duration 3:27)
View the video lesson, take notes and complete the problems below
Check solutions
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YOU TRY
Solve for π₯π₯ and check solutions
a) β2π₯π₯ + 1 β βπ₯π₯ = 1
Check solutions
b) β2π₯π₯ + 6 β βπ₯π₯ + 4 = 1
Check solutions
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C. RADICAL EQUATIONS WITH HIGHER ROOTS
MEDIA LESSON Solve equations with radicals β odd roots (Duration 2:42)
View the video lesson, take notes and complete the problems below
The opposite of taking a root is to do an ______________________________.
βπ₯π₯3 = 4 then π₯π₯ =_______
Example: a) β2π₯π₯ β 53 = 6 b) β4π₯π₯ β 75 = 2
YOU TRY
Solve for ππ.
a) βππ β 13 = β4
b) βπ₯π₯2 β 6π₯π₯4 = 2
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EXERCISE Solve. Be sure to verify all solutions.
1) β2π₯π₯ + 3 β 3 = 0
2) β6π₯π₯ β 5 β π₯π₯ = 0
3) 3 + π₯π₯ = β6π₯π₯ + 13
4) β3 β 3π₯π₯ β 1 = 2π₯π₯
5) β4π₯π₯ + 5 β βπ₯π₯ + 4 = 2
6) β2π₯π₯ + 4 β βπ₯π₯ + 3 = 1
7) β2π₯π₯ + 6 β βπ₯π₯ + 4 = 1
8) β6 β 2π₯π₯ β β2π₯π₯ + 3 = 3
9) β5π₯π₯ + 1 β 4 = 0 10) βπ₯π₯ + 1 = βπ₯π₯ + 1
11) π₯π₯ β 1 = β7 β π₯π₯
12) β2π₯π₯ + 2 = 3 + β2π₯π₯ β 1
13) β3π₯π₯ + 4 β βπ₯π₯ + 2 = 2
14) β7π₯π₯ + 2 β β3π₯π₯ + 6 = 6
15) β4π₯π₯ β 3 = β3π₯π₯ + 1 + 1
16) βπ₯π₯ + 2 β βπ₯π₯ = 2
17) βπ₯π₯ + 25 = ββ35
18) β5π₯π₯ + 13 β 2 = 4
19) 3βπ₯π₯3 = 12
20) β7π₯π₯ + 153 = 1
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CHAPTER REVIEW KEY TERMS AND CONCEPTS
Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson.
Radicals
Radicand
Like-radicals
Product rule for radicals
Rationalize denominator process
Conjugates
To rationalize the denominator with square roots
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