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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 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

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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 ⋅ 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

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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.

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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

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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

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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

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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 = _____________________________________________________________

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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

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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

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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

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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

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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 −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

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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

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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 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

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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

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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�

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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: (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�

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D. MULTIPLY RADICALS WITH SPECIAL-PRODUCT FORMULAS

MEDIA LESSON Multiply radicals using the perfect square formula (Duration 3:44)

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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)

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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� = ____________________________________________________________

= ____________________________________________________________

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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

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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) �100𝑥𝑥5𝑥𝑥

, assume 𝑥𝑥 is positive

b) �64𝑥𝑥2𝑦𝑦53

�4𝑦𝑦23 , assume 𝑦𝑦 is not 0

MEDIA LESSON Divide expressions with radicals (Duration 4:20 )

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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

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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

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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)

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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.,

𝟏𝟏√𝒙𝒙

∙√𝒙𝒙√𝒙𝒙

MEDIA LESSON Rationalize monomials (Duration 3:42)

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Example: Simplify by rationalizing the denominator. a) 20

√10 b) 35

3√7

MEDIA LESSON Rationalize monomials with variables (Duration 4:58)

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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

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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 – 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

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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:

(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)

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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

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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) √2

4+√10

YOU TRY

Simplify.

a) 2

√3−5

b) 3−√52−√3

c) 2√5−3√75√6+4√2

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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

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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 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)

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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

= ____________

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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

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B. RADICAL EQUATIONS WITH TWO SQUARE ROOTS

MEDIA LESSON Solve equations with two radicals (Duration 5:11)

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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

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349

MEDIA LESSON Solve equations with two radicals – part 2 (Duration 4:33 )

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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

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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

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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

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