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Chapter 12
317
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 𝒏𝒏𝒏𝒏𝒏𝒏 ROOT .................................................... 322
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏𝒏𝒏𝒏𝒏 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
318
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
Chapter 12
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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 ⋅ 1 = ___________________ 62 = 6 ⋅ 6 =___________________
22 = 2 ⋅ 2 = ___________________ 72 = 7 ⋅ 7 = ___________________
32 = 3 ⋅ 3 = ___________________ 82 = 8 ⋅ 8 =___________________
42 = 4 ⋅ 4 = ___________________ 92 = 9 ⋅ 9 =___________________
52 = 5 ⋅ 5 = ___________________ 102 = 10 ⋅ 10 =___________________
To determine the square root of a number, we have a special symbol.
√9
The square root of a number is the number times itself that equals the given number.
√9 = ____________________________________________________________
√36 = ____________________________________________________________
√49 = ____________________________________________________________
√81 =____________________________________________________________
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 ⋅ 5 = 25 and −5 ∙ −5 = 25
To avoid confusion
√25 = 5 and −√25 = −5
What about these square roots?
√20
√61
Chapter 12
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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:
√441 = _______________ √484 =_______________
√529 =_______________
√576 =_______________ √625 =_______________ √676 =_______________ √729 =_______________
√784 =_______________ √841 = _______________
√900 = _______________
MEDIA LESSON Principal nth square roots vs. general square roots (Duration 5:23 )
Note: In this class, we will only consider the principal 𝒏𝒏𝒏𝒏𝒏𝒏roots 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 “square” of a number, we “square root” a number; instead of the “cube” of a number, we “cube root” a number to reverse the square to find the base. Square roots are the most common type of radical used in algebra.
Definition
If 𝒂𝒂 is a positive real number, then the principal square root of a number 𝒂𝒂 is defined as
√𝒂𝒂 = 𝒃𝒃 if and only if 𝒂𝒂 = 𝒃𝒃𝟐𝟐
The √ is the radical symbol, and 𝒂𝒂 is called the radicand.
If given something like √𝒂𝒂𝟑𝟑, then 3 is called the root or index; hence, √𝒂𝒂 𝟑𝟑
is called the cube root or third root of 𝒂𝒂. In general,
√𝒂𝒂𝒏𝒏 = 𝒃𝒃 if and only if 𝒂𝒂 = 𝒃𝒃𝒏𝒏
If 𝒏𝒏 is even, then 𝒂𝒂 and 𝒃𝒃 must be greater than or equal to zero. If 𝒏𝒏 is odd, then 𝒂𝒂 and 𝒃𝒃 must be any real number.
Here are some examples of principal square roots:
√1 = 1 √121 = 11 √4 = 2 √625 = 25 √9 = 3 √−81 is not a real number
The final example √−81 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 −81 < 0, then there is no real number in which we can square and will result in −81,i.e., ?2 = −81. 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.
Chapter 12
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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 𝒏𝒏𝒏𝒏𝒏𝒏 root of 𝒂𝒂 is the 𝒏𝒏𝒏𝒏𝒏𝒏 root that has the same sign as 𝒂𝒂, and it is denoted by the radical symbol.
√𝒂𝒂𝒏𝒏 We read this as the “___________________________”, “______________”, or “_______________”. The positive integer ______________________________ of the radical. If 𝑛𝑛 = 2, ____________ the index.
The number _______________________.
√4 =________________
√164 =_______________
−√4 =________________
−√164 =_______________
Square roots (n = 2) √1 =________________________________ −√1 =________________________________
√4 = ________________________________ −√4 = ________________________________
√9 = ________________________________ −√9 = ________________________________
√16 = _______________________________ −√16 = _______________________________
√25 = _______________________________ −√25 = _______________________________
Cube roots (n = 3)
√13 = __________________________ √−13 = __________________________
√83 = __________________________ √−83 = __________________________
√273 =__________________________ √−273 =_________________________
√643 = __________________________ √−643 = _________________________
√1253 =_________________________ √−1253 =________________________
Example: Simplify
1) √36 =
2) −√81 =
3) �49 =
4) √643 =
5) √325 = 6) − √−83 =
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Inverse properties of 𝒏𝒏𝒏𝒏𝒏𝒏 Powers and 𝒏𝒏𝒏𝒏𝒏𝒏 Roots
If 𝒂𝒂 has a principal 𝒏𝒏𝒏𝒏𝒏𝒏 root, then____________________________.
If 𝒏𝒏 is odd, then ______________________________. If 𝒏𝒏 is even, then ______________________________. We need the ____________________________ for any 𝒏𝒏𝒏𝒏𝒏𝒏 root with an _____________ exponent
for which the index is ____________ to assure the 𝒏𝒏𝒏𝒏𝒏𝒏 root is ______________.
Example: Simplify
1) √𝑥𝑥2
2) √𝑥𝑥93
3) √𝑥𝑥84
4) �𝑦𝑦124
C. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏𝒏𝒏𝒏𝒏 ROOT
MEDIA LESSON Simplify perfect 𝒏𝒏𝒏𝒏𝒏𝒏roots (Duration 4:04 )
View the video lesson, take notes and complete the problems below
Example: a) √81
b) √273
c) √164
d) √243
Chapter 12
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MEDIA LESSON Simplify perfect 𝒏𝒏𝒏𝒏𝒏𝒏roots – negative radicands (Duration 4:32 )
View the video lesson, take notes and complete the problems below
Example: Simplify each of the following.
a) √164 = ________________________________________________________________________
b) √−325 = ________________________________________________________________________
c) √−646 = ________________________________________________________________________
YOU TRY Simplify. Show your work.
a) √−36
b) √−64 3
c) − √6254
d) √15
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏𝒏𝒏𝒏𝒏 ROOT USING EXPONENT RULE
There is a more efficient way to find the 𝑛𝑛𝑡𝑡ℎ root by using the exponent rule but first let’s 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 “stacked division” method.
MEDIA LESSON Prime factorization – stacked division method (Duration 3:45)
View the video lesson, take notes and complete the problems below
a) 1,350 b) 168
Chapter 12
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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: √𝒎𝒎𝒏𝒏 where 𝒏𝒏 is the _______________
Roots of an expression with exponents: _________________the ________________ by the __________.
Example: Simplify.
a) �46,656 = b) �1,889,5685 =
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) √𝑧𝑧93
b) √𝑚𝑚6
c) −√𝑛𝑛105
YOU TRY Simplify the following radicals using the exponent rule. Show your work.
a) √646
b) √7293
c) �𝑥𝑥2𝑦𝑦4𝑧𝑧10
d) �𝑥𝑥21𝑦𝑦427
Chapter 12
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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 √8 in a calculator, the calculator would display
2.828427124746190097603377448419… 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 – no decimals. When we say to simplify an expression with radicals, the simplified expression should have
• a radical, unless the radical reduces to an integer • a radicand with no factors containing perfect squares • no decimals
Following these guidelines ensures the expression is in its simplest form.
Product rule for radicals
If 𝒂𝒂,𝒃𝒃 are any two positive real numbers, then
√𝑎𝑎𝑎𝑎 = √𝑎𝑎 ∙ √𝑎𝑎 In general, if 𝒂𝒂,𝒃𝒃 are any two positive real numbers, then
√𝑎𝑎𝑎𝑎𝑛𝑛 = √𝑎𝑎𝑛𝑛 ∙ √𝑎𝑎𝑛𝑛
Where 𝒏𝒏 is a positive integer and 𝒏𝒏 ≥ 𝟐𝟐.
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, 𝒂𝒂: √𝒂𝒂𝟐𝟐 = 𝒂𝒂
For example: √25 = √5 ⋅ 5 = √52 = 5 The product rule for square roots
Given that 𝑎𝑎 and 𝑎𝑎 are non-negative real numbers, ___________________________________________.
√45 = ________________________________________________________________________. Example: √8 = _____________________________________________________________
√48 = _____________________________________________________________
√150 = _____________________________________________________________
�1,350 = _____________________________________________________________
Chapter 12
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MEDIA LESSON Simplify radicals with not perfect radicants – 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
√𝑎𝑎2𝑎𝑎 =
√𝑎𝑎𝑛𝑛𝑎𝑎𝑛𝑛 = When we divide if there is a remainder, the remainder ________________________________________.
Example:
a) √72 b) √7503
YOU TRY
Simplify. Show your work.
a) √75
b) √2003
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) −8√600 b) 3 √−965
YOU TRY
Simplify. a) 5√63
b) −8√392
Chapter 12
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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) √𝑎𝑎13𝑎𝑎23𝑐𝑐10𝑑𝑑34
b) �125𝑥𝑥4𝑦𝑦𝑧𝑧5
YOU TRY
Simplify. Assume all variables are positive. a) �𝑥𝑥6𝑦𝑦5
b) −5�18𝑥𝑥4𝑦𝑦6𝑧𝑧10
c) �20𝑥𝑥5𝑦𝑦9𝑧𝑧6
Chapter 12
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EXERCISE Simplify. Show all your work. Assume all variables are positive.
1) √245
2) √36
3) √12
4) 3√12
5) 6√128
6) −8√392
7) √192𝑛𝑛
8) √196𝑣𝑣2
9) √252𝑥𝑥2
10) −√100𝑘𝑘4
11) −7√64𝑥𝑥4
12) −5√36𝑚𝑚
13) −4�175𝑝𝑝4
14) 8�112𝑝𝑝2
15) −2√128𝑛𝑛
16) �45𝑥𝑥2𝑦𝑦2
17) �16𝑥𝑥3𝑦𝑦3
18) �320𝑥𝑥4𝑦𝑦4
19) −�32𝑥𝑥𝑦𝑦2𝑧𝑧3
20) 5�245𝑥𝑥2𝑦𝑦3
21) −2√180𝑢𝑢3𝑣𝑣
22) √72𝑎𝑎3𝑎𝑎4
23) 2�80ℎ𝑗𝑗4𝑘𝑘
24) 6√50𝑎𝑎4𝑎𝑎𝑐𝑐2
25) 8√98𝑚𝑚𝑛𝑛
26) √512𝑎𝑎4𝑎𝑎2
27) √100𝑚𝑚4𝑛𝑛3
28) −8�180𝑥𝑥4𝑦𝑦2𝑧𝑧4
29) 2�72𝑥𝑥2𝑦𝑦2
30) −5�36𝑥𝑥3𝑦𝑦4
Simplify. Show all your work. Assume all variables are positive.
31) √6253
32) √7503
33) √8753
34) −4 √964
35) 6 √1124
36) √648𝑎𝑎24
37) √224𝑛𝑛35
38) �224𝑝𝑝55
39) −3 √896𝑟𝑟7
40) −2 √−48𝑣𝑣73
41) −7 √320𝑛𝑛63
42) �−135𝑥𝑥5𝑦𝑦33
43) �−32𝑥𝑥4𝑦𝑦43
44) �256𝑥𝑥4𝑦𝑦63
45) 7 �−81𝑥𝑥3𝑦𝑦73
46) 2 √375𝑢𝑢2𝑣𝑣83
47) −3 √192𝑎𝑎𝑎𝑎23
48) 6 �−54𝑚𝑚8𝑛𝑛3𝑝𝑝73
49) 6 �648𝑥𝑥5𝑦𝑦7𝑧𝑧24 50) 9�9𝑥𝑥2𝑦𝑦5𝑧𝑧3
Chapter 12
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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. Let’s look at an example with like terms and like radicals.
2𝑥𝑥 + 5𝑥𝑥 (2 + 5)𝑥𝑥
7𝑥𝑥
2√3 + 5√3 (2 + 5)√3
7√3 Notice that when we combined the terms with √3, it was similar to combining terms with 𝑥𝑥. 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
𝒂𝒂√𝒙𝒙 ± 𝒃𝒃√𝒙𝒙 = (𝒂𝒂 ± 𝒃𝒃)√𝒙𝒙,
Where 𝒂𝒂,𝒃𝒃 are real numbers and 𝒙𝒙 is some positive real number.
In general, for any root 𝒏𝒏, 𝒂𝒂√𝒙𝒙𝒏𝒏 ± 𝒃𝒃√𝒙𝒙𝒏𝒏 = (𝒂𝒂 ± 𝒃𝒃)√𝒙𝒙𝒏𝒏 ,
Where 𝒂𝒂,𝒃𝒃 are real numbers and 𝒙𝒙 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: 2𝑥𝑥 − 5𝑦𝑦 + 3𝑥𝑥 + 2𝑦𝑦
_______________________
Simplify: 2√3 − 5√7 + 3√3 + 2√7
_______________________
When adding and subtracting radicals, we can ______________________________________________. Example:
a) −4√6 + 2√11 + √11 − 5√6 b) √53 + 3√5 − 8√53 + 2√5
YOU TRY
Simplify
a) 7√65 + 4√35 − 9√35 + √65
b) −3√2 + 3√5 + 3√5
Chapter 12
330
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 −2�50𝑥𝑥5 + 5�18𝑥𝑥5 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) 2√18 + √50 b) 𝑥𝑥 �𝑥𝑥2𝑦𝑦53 + 𝑦𝑦 �𝑥𝑥5𝑦𝑦23
YOU TRY
Simplify.
a) 5√45 + 6√18 − 2√98 + √20
b) 4√543 − 9√163 + 5√93
Chapter 12
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EXERCISE Simplify. In this section, we assume all variables to be positive.
1) 2√5 + 2√5 + 2√5
2) −2√6 − 2√6 −√6
3) 3√6 + 3√5 + 2√5
4) 2√2 − 3√18 − √2
5) 3√2 + 2√8 − 3√18
6) −3√6 −√12 + 3√3
7) 3√18 − √2 − 3√2
8) −2√18− 3√8 − √20 + 2√20
9) −2√24− 2√6 + 2√6 + 2√20
10) 3√24 − 3√27 + 2√6 + 2√8
11) −2√163 + 2√163 + 2√23
12) 2√2434 − 2√2434 − √34
13) √6254 -5√6254 + √643 − 5√643
14) 3√24 − 2√24 − √2434
15) −√3244 + 3√3244 − 3√44
16) 2√24 + 2√34 + 3√644 − √34
17) −3√65 − √645 + 2√1925 − 2√645
18) 2√1605 − 2√1925 − √1605 − √−1605
19) −√2566 − 2√46 − 3√3206 − 2√1286
20) 3√1353 − √813 − √1353
21) −3√18𝑥𝑥5 − √8𝑥𝑥5 + 2√8𝑥𝑥5 + 2√8𝑥𝑥5
22) −2�2𝑥𝑥𝑦𝑦 − �2𝑥𝑥𝑦𝑦 + 3�8𝑥𝑥𝑦𝑦 + 3�8𝑥𝑥𝑦𝑦
23) 2√6𝑥𝑥2 − √54𝑥𝑥2 − 3�27𝑥𝑥2𝑦𝑦 − �3𝑥𝑥2𝑦𝑦
24) 2𝑥𝑥�20𝑦𝑦2 + 7𝑦𝑦√20𝑥𝑥2 − �3𝑥𝑥𝑦𝑦
25) 3√24𝑡𝑡 − 3√54𝑡𝑡 − 2√96𝑡𝑡 + 2√150𝑡𝑡
Chapter 12
332
SECTION 12.3: MULTIPLY AND DIVIDE RADICALS
Recall the product rule for radicals in the previous section:
Product rule for radicals
If 𝒂𝒂,𝒃𝒃 are any two positive real numbers, then
√𝑎𝑎𝑎𝑎 = √𝑎𝑎 ∙ √𝑎𝑎 In general, if 𝒂𝒂,𝒃𝒃 are any two positive real numbers, then
√𝑎𝑎𝑎𝑎𝑛𝑛 = √𝑎𝑎𝑛𝑛 ∙ √𝑎𝑎𝑛𝑛
Where 𝒏𝒏 is a positive integer and 𝒏𝒏 ≥ 𝟐𝟐.
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): p√𝑢𝑢𝑛𝑛 ⋅ 𝑞𝑞 √𝑣𝑣𝑛𝑛 = ________________
Example 1: √2 ⋅ √3 = ______________________________________
Example 2: 3√53 ⋅ 4√73 = ____________________________________
Multiply:
a) √15 ⋅ √6 b) √183 ⋅ √603
c) 3√12 ⋅ 5√63
d) −2√404 ⋅ 7√184
e) −√6 · −3√6
Chapter 12
333
YOU TRY
Simplify: a) −5√14 ∙ 4√6
b) 2 √183 ∙ 6 √153
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) √18𝑥𝑥3 ⋅ √30𝑥𝑥2 b) √16𝑥𝑥23 ⋅ √81𝑥𝑥23
YOU TRY
Simplify.
a) √8𝑥𝑥25 ∙ √4𝑥𝑥35
b) √60𝑥𝑥4 ∙ √6𝑥𝑥7
Chapter 12
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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: √7�√14 − √2� √3�5 + √3�
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) √𝑥𝑥�2√𝑥𝑥 − 3�
b) 4�𝑦𝑦�5�𝑥𝑥𝑦𝑦3 − �𝑦𝑦3�
c) √𝑧𝑧3 �√𝑧𝑧23 − 7√𝑧𝑧53 + 2 √𝑧𝑧83 �
YOU TRY
Simplify.
a) 7√6 (3√10 − 5√15)
b) √3�7√15𝑥𝑥3 + 8𝑥𝑥√60𝑥𝑥�
Chapter 12
335
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: (𝑎𝑎 + 𝑎𝑎)(𝑐𝑐 + 𝑎𝑎) = ____________________________________
Always be sure your final answer is ____________________________.
Example: a) �3√7 − 2√5��√7 + 6√5�
b) �2 √93 + 5� �4 √33 − 1�
MEDIA LESSON Multiply binomials with radicals with variables (Duration 5:29)
View the video lesson, take notes and complete the problems below
Example: a) �2√𝑥𝑥 + 3��5√𝑥𝑥 − 4� b) �3𝑥𝑥2 + √𝑥𝑥23 � �2 √𝑥𝑥3 − 1�
YOU TRY
Simplify.
a) (√5 − 2√3)(4√10 + 6√6)
b) �3√𝑣𝑣 + 2√3��5√𝑣𝑣 − 7√3�
Chapter 12
336
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: (𝑎𝑎 + 𝑎𝑎)2 = ________________________________
Always be sure your final answer is _________________________
Example:
a) �√6 −√2�2
b) �2 + 3√7�2
Conjugates
Recall the Difference of for two squares formula: (𝒂𝒂 − 𝒃𝒃)(𝒂𝒂 + 𝒃𝒃) = 𝒂𝒂𝟐𝟐 − 𝒃𝒃𝟐𝟐 Notice in the 2 factors (𝒂𝒂 − 𝒃𝒃) and (𝒂𝒂 + 𝒃𝒃) 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 − 5 3 + 5 𝑥𝑥 + 5 𝑥𝑥 − 5
1 − √2 1 + √2 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: (𝑎𝑎 − 𝑎𝑎)(𝑎𝑎 + 𝑎𝑎) = ____________________________________
�3 − √6��3 + √6� = ____________________________________________________________________
�√2 −√5��√2 + √5� = _________________________________________________________________
�2√3 + 3√7��2√3 − 3√7� = ____________________________________________________________
= ____________________________________________________________
Chapter 12
337
YOU TRY
a) Simplify: (5√7 + √2)2
b) Simplify: (8 −√5)(8 + √5)
E. SIMPLIFY QUOTIENTS WITH RADICALS
Quotient rule for radicals
If 𝒂𝒂,𝒃𝒃 are any two positive real numbers, where 𝒃𝒃 ≠ 𝟎𝟎, then
�𝒂𝒂𝒃𝒃
=√𝒂𝒂√𝒃𝒃
If 𝒂𝒂,𝒃𝒃 are any two positive real numbers, where 𝒃𝒃 ≠ 𝟎𝟎, then
�𝒂𝒂𝒃𝒃
𝒏𝒏=√𝒂𝒂𝒏𝒏
√𝒃𝒃𝒏𝒏
Where 𝒏𝒏 is a positive integer and 𝒏𝒏 ≥ 𝟐𝟐.
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) �7516
b) �3244
3
Chapter 12
338
MEDIA LESSON Divide radicals with variables (Duration 4:34 )
View the video lesson, take notes and complete the problems below
Examples:
a) �100𝑥𝑥5𝑥𝑥
, assume 𝑥𝑥 is positive
b) �64𝑥𝑥2𝑦𝑦53
�4𝑦𝑦23 , assume 𝑦𝑦 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 + √175
10
b) 8 − √48
6
YOU TRY
Simplify.
a) −3+√27
3
b) �44𝑦𝑦6𝑎𝑎4
�9𝑦𝑦2𝑎𝑎8
c) 15 √1083
20 √23
Chapter 12
339
EXERCISE Simplify. Assume all variables are positive.
1) −4√16 ∙ 3√5
2) 3√10 ∙ √20
3) −5√10𝑟𝑟2 ∙ √5𝑟𝑟3
4) √12𝑚𝑚 ∙ √15𝑚𝑚
5) 3√4𝑎𝑎43 ∙ √10𝑎𝑎33
6) √4𝑥𝑥33 ∙ √2𝑥𝑥43
7) √6(√2 + 2)
8) 5√10(5𝑛𝑛 + √2)
9) −5√15(3√3 + 2)
10) 5√15(3√3 + 2)
11) √10(√5 + √2)
12) √15(√5 − 3√3𝑣𝑣)
13) (2 + 2�2)(−3 + √2)
14) (−2 + √3)(−5 + 2√3)
15) (−5 − 4√3)(−3− 4√3)
16) (√5 − 5)(2√5 − 1)
17) (√2𝑎𝑎 + 2√3𝑎𝑎)(3√2𝑎𝑎 + √5𝑎𝑎)
18) (5√2 − 1)(−√2𝑚𝑚 + 5)
19) √10√6
20) √5
4√125
21) √125√100
22) √53
4 √43
23) 2√43√3
24) 3 √103
5 √273
25) �12𝑝𝑝2
�3𝑝𝑝
26) 4+ 8√452√4
27) 3+ √12√3
28) 4−2√23√32
Chapter 12
340
29) 4−√30√15
30) 5 √5𝑟𝑟44
√8𝑟𝑟24
31) 5𝑥𝑥2
4𝑥𝑥𝑦𝑦�3𝑥𝑥𝑦𝑦 32) (5 + 2√6)2
33) (𝑥𝑥 − 𝑥𝑥√5)2 34) (√3 − √7)2
35) (5√6 + 2√3)2
36) (√2 − √5)(√2 + √5)
37) (√𝑥𝑥 − �𝑦𝑦)(√𝑥𝑥 + �𝑦𝑦)
38) (4 − 2√3)(4 + 2√3)
39) (𝑥𝑥 − 𝑦𝑦√3)(𝑥𝑥 + 𝑦𝑦√3)
40) (9√𝑥𝑥 + �𝑦𝑦)(9√𝑥𝑥 − �𝑦𝑦)
Chapter 12
341
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.,
𝟏𝟏√𝒙𝒙
∙√𝒙𝒙√𝒙𝒙
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
√10 b) 35
3√7
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) √7𝑎𝑎𝑎𝑎√6𝑎𝑎𝑐𝑐2
b) � 5𝑥𝑥𝑦𝑦3
15𝑥𝑥𝑦𝑦𝑥𝑥
YOU TRY
Simplify.
a) √6√5
b) 6√1412√22
c) √3−92√6
Chapter 12
342
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 – Monomial higher root
Use the ____________________
To clear radicals _____________ by extra needed factors in denominator (multiply by the same on top!)
Hint: ___________________ numbers!
Example:
a) 5√𝑎𝑎27
b) � 79𝑎𝑎2𝑎𝑎
3
YOU TRY
Simplify.
a) 4 √23
7 √253
b) 3 √114
√24
Chapter 12
343
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:
(𝑎𝑎 + 𝑎𝑎)(𝑎𝑎 − 𝑎𝑎) = 𝑎𝑎2 − 𝑎𝑎2
Rationalize denominators using the conjugate
We rationalize denominators of the type 𝑎𝑎 ± √𝑎𝑎 by multiplying the numerator and denominator by their conjugates, e.g.,
1𝑎𝑎 + √𝑎𝑎
∙𝑎𝑎 − √𝑎𝑎𝑎𝑎 − √𝑎𝑎
=𝑎𝑎 − √𝑎𝑎
(𝑎𝑎)2 − (√𝑎𝑎)2
The conjugate for • 𝑎𝑎 + √𝑎𝑎 is 𝑎𝑎 − √𝑎𝑎 • 𝑎𝑎 − √𝑎𝑎 is 𝑎𝑎 + √𝑎𝑎
The case is similar for when there is something like √𝑎𝑎 ± √𝑎𝑎 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 – Binomials
What doesn’t work: 1
2+√3
Recall: �2 + √3� _______________________
Multiply by the ________________________
Example:
a) 6
5−√3 b)
3−5√24+2√2
Chapter 12
344
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) √2
4+√10
YOU TRY
Simplify.
a) 2
√3−5
b) 3−√52−√3
c) 2√5−3√75√6+4√2
Chapter 12
345
EXERCISE Simplify. Assume all variables are positive.
1) 2√43√3
2) √12√3
3) √23√5
4) 4√3√15
5) 4+2√3√9
6) √53
4 √43
7) 2√23 8)
6 √23
√93 9) 8
√3𝑥𝑥23
10) 2𝑥𝑥√𝑥𝑥3 11)
𝑣𝑣√2𝑣𝑣34 12)
1√5𝑥𝑥4
13) 4+2√35√4
14) 2−5√54√13
15) √2−3√3
√3
16) 5
3√5+√2 17)
25+√2
18) 3
4−3√3
19) 4
3+√5 20) − 4
4−4√2 21)
45 + √5𝑥𝑥2
22) 5
2+√5𝑟𝑟3 23)
2−√5−3+√5
24) √3+√22√3−√2
25) 4√2+33√2+√3
26) 5
√3+4√5 27)
2√5+√31−√3
28) 𝑎𝑎−𝑎𝑎
√𝑎𝑎−√𝑎𝑎 29)
7√𝑎𝑎+√𝑎𝑎
30) 𝑎𝑎−√𝑎𝑎𝑎𝑎+√𝑎𝑎
Chapter 12
346
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 𝒙𝒙 ≥ 𝟎𝟎 and 𝒂𝒂 ≥ 𝟎𝟎, then
√𝒙𝒙 = 𝒂𝒂 if and only if 𝒙𝒙 = 𝒂𝒂𝟐𝟐 If 𝒙𝒙 ≥ 𝟎𝟎 and 𝒂𝒂 is a real number, then
√𝒙𝒙𝒏𝒏 = 𝒂𝒂 if and only if 𝒙𝒙 = 𝒂𝒂𝒏𝒏 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.
�√𝑥𝑥�2
= ___________ �√𝑥𝑥3 �3
= ____________
Chapter 12
347
Example: Solve. a) √𝑥𝑥 − 7 = 11
b) √3𝑥𝑥 + 2 − 7 = 0
c) 2√5𝑥𝑥 − 13 − 8 = 0
d) √𝑥𝑥 + 6 = 𝑥𝑥
YOU TRY
Solve for 𝑥𝑥.
a) √7𝑥𝑥 + 2 = 4
b) √𝑥𝑥 + 3 = 5
c) 𝑥𝑥 + √4𝑥𝑥 + 1 = 5
d) √𝑥𝑥 + 6 = 𝑥𝑥 + 4
Chapter 12
348
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) √2𝑥𝑥 + 3 − √𝑥𝑥 − 8 = 0 b) 3 + √𝑥𝑥 − 6 = √𝑥𝑥 + 9
Chapter 12
349
MEDIA LESSON Solve equations with two radicals – part 2 (Duration 4:33 )
View the video lesson, take notes and complete the problems below
Example: Solve the equation. √1 − 8𝑥𝑥 − √−16𝑥𝑥 − 12 = 1
MEDIA LESSON Solve equations with two radicals – part 3 – check solutions (Duration 3:27)
View the video lesson, take notes and complete the problems below
Check solutions
Chapter 12
350
YOU TRY
Solve for 𝑥𝑥 and check solutions
a) √2𝑥𝑥 + 1 − √𝑥𝑥 = 1
Check solutions
b) √2𝑥𝑥 + 6 − √𝑥𝑥 + 4 = 1
Check solutions
Chapter 12
351
C. RADICAL EQUATIONS WITH HIGHER ROOTS
MEDIA LESSON Solve equations with radicals – 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 ______________________________.
√𝑥𝑥3 = 4 then 𝑥𝑥 =_______
Example: a) √2𝑥𝑥 − 53 = 6 b) √4𝑥𝑥 − 75 = 2
YOU TRY
Solve for 𝑛𝑛.
a) √𝑛𝑛 − 13 = −4
b) √𝑥𝑥2 − 6𝑥𝑥4 = 2
Chapter 12
352
EXERCISE Solve. Be sure to verify all solutions.
1) √2𝑥𝑥 + 3 − 3 = 0
2) √6𝑥𝑥 − 5 − 𝑥𝑥 = 0
3) 3 + 𝑥𝑥 = √6𝑥𝑥 + 13
4) √3 − 3𝑥𝑥 − 1 = 2𝑥𝑥
5) √4𝑥𝑥 + 5 − √𝑥𝑥 + 4 = 2
6) √2𝑥𝑥 + 4 − √𝑥𝑥 + 3 = 1
7) √2𝑥𝑥 + 6 − √𝑥𝑥 + 4 = 1
8) √6 − 2𝑥𝑥 − √2𝑥𝑥 + 3 = 3
9) √5𝑥𝑥 + 1 − 4 = 0 10) √𝑥𝑥 + 1 = √𝑥𝑥 + 1
11) 𝑥𝑥 − 1 = √7 − 𝑥𝑥
12) √2𝑥𝑥 + 2 = 3 + √2𝑥𝑥 − 1
13) √3𝑥𝑥 + 4 − √𝑥𝑥 + 2 = 2
14) √7𝑥𝑥 + 2 − √3𝑥𝑥 + 6 = 6
15) √4𝑥𝑥 − 3 = √3𝑥𝑥 + 1 + 1
16) √𝑥𝑥 + 2 − √𝑥𝑥 = 2
17) √𝑥𝑥 + 25 = √−35
18) √5𝑥𝑥 + 13 − 2 = 4
19) 3√𝑥𝑥3 = 12
20) √7𝑥𝑥 + 153 = 1
Chapter 12
353
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
Chapter 12
354
