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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 ๐๐๐๐๐๐ ROOT .................................................... 322
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ๐๐๐๐๐๐ 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
319
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 โ 1 = ___________________ 62 = 6 โ 6 =___________________
22 = 2 โ 2 = ___________________ 72 = 7 โ 7 = ___________________
32 = 3 โ 3 = ___________________ 82 = 8 โ 8 =___________________
42 = 4 โ 4 = ___________________ 92 = 9 โ 9 =___________________
52 = 5 โ 5 = ___________________ 102 = 10 โ 10 =___________________
To determine the square root of a number, we have a special symbol.
โ9
The square root of a number is the number times itself that equals the given number.
โ9 = ____________________________________________________________
โ36 = ____________________________________________________________
โ49 = ____________________________________________________________
โ81 =____________________________________________________________
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 โ 5 = 25 and โ5 โ โ5 = 25
To avoid confusion
โ25 = 5 and โโ25 = โ5
What about these square roots?
โ20
โ61
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:
โ441 = _______________ โ484 =_______________
โ529 =_______________
โ576 =_______________ โ625 =_______________ โ676 =_______________ โ729 =_______________
โ784 =_______________ โ841 = _______________
โ900 = _______________
MEDIA LESSON Principal nth square roots vs. general square roots (Duration 5:23 )
Note: In this class, we will only consider the principal ๐๐๐๐๐๐roots 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 โsquareโ of a number, we โsquare rootโ a number; instead of the โcubeโ of a number, we โcube rootโ a number to reverse the square to find the base. Square roots are the most common type of radical used in algebra.
Definition
If ๐๐ is a positive real number, then the principal square root of a number ๐๐ is defined as
โ๐๐ = ๐๐ if and only if ๐๐ = ๐๐๐๐
The โ is the radical symbol, and ๐๐ is called the radicand.
If given something like โ๐๐๐๐, then 3 is called the root or index; hence, โ๐๐ ๐๐
is called the cube root or third root of ๐๐. In general,
โ๐๐๐๐ = ๐๐ if and only if ๐๐ = ๐๐๐๐
If ๐๐ is even, then ๐๐ and ๐๐ must be greater than or equal to zero. If ๐๐ is odd, then ๐๐ and ๐๐ must be any real number.
Here are some examples of principal square roots:
โ1 = 1 โ121 = 11 โ4 = 2 โ625 = 25 โ9 = 3 โโ81 is not a real number
The final example โโ81 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 โ81 < 0, then there is no real number in which we can square and will result in โ81,i.e., ?2 = โ81. 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 ๐๐๐๐๐๐ root of ๐๐ is the ๐๐๐๐๐๐ root that has the same sign as ๐๐, and it is denoted by the radical symbol.
โ๐๐๐๐ We read this as the โ___________________________โ, โ______________โ, or โ_______________โ. The positive integer ______________________________ of the radical. If ๐๐ = 2, ____________ the index.
The number _______________________.
โ4 =________________
โ164 =_______________
โโ4 =________________
โโ164 =_______________
Square roots (n = 2) โ1 =________________________________ โโ1 =________________________________
โ4 = ________________________________ โโ4 = ________________________________
โ9 = ________________________________ โโ9 = ________________________________
โ16 = _______________________________ โโ16 = _______________________________
โ25 = _______________________________ โโ25 = _______________________________
Cube roots (n = 3)
โ13 = __________________________ โโ13 = __________________________
โ83 = __________________________ โโ83 = __________________________
โ273 =__________________________ โโ273 =_________________________
โ643 = __________________________ โโ643 = _________________________
โ1253 =_________________________ โโ1253 =________________________
Example: Simplify
1) โ36 =
2) โโ81 =
3) ๏ฟฝ49 =
4) โ643 =
5) โ325 = 6) โ โโ83 =
Chapter 12
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Inverse properties of ๐๐๐๐๐๐ Powers and ๐๐๐๐๐๐ Roots
If ๐๐ has a principal ๐๐๐๐๐๐ root, then____________________________.
If ๐๐ is odd, then ______________________________. If ๐๐ is even, then ______________________________. We need the ____________________________ for any ๐๐๐๐๐๐ root with an _____________ exponent
for which the index is ____________ to assure the ๐๐๐๐๐๐ root is ______________.
Example: Simplify
1) โ๐ฅ๐ฅ2
2) โ๐ฅ๐ฅ93
3) โ๐ฅ๐ฅ84
4) ๏ฟฝ๐ฆ๐ฆ124
C. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ๐๐๐๐๐๐ ROOT
MEDIA LESSON Simplify perfect ๐๐๐๐๐๐roots (Duration 4:04 )
View the video lesson, take notes and complete the problems below
Example: a) โ81
b) โ273
c) โ164
d) โ243
Chapter 12
323
MEDIA LESSON Simplify perfect ๐๐๐๐๐๐roots โ negative radicands (Duration 4:32 )
View the video lesson, take notes and complete the problems below
Example: Simplify each of the following.
a) โ164 = ________________________________________________________________________
b) โโ325 = ________________________________________________________________________
c) โโ646 = ________________________________________________________________________
YOU TRY Simplify. Show your work.
a) โโ36
b) โโ64 3
c) โ โ6254
d) โ15
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL ๐๐๐๐๐๐ ROOT USING EXPONENT RULE
There is a more efficient way to find the ๐๐๐ก๐กโ root by using the exponent rule but first letโs 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 โstacked divisionโ method.
MEDIA LESSON Prime factorization โ 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: โ๐๐๐๐ where ๐๐ is the _______________
Roots of an expression with exponents: _________________the ________________ by the __________.
Example: Simplify.
a) ๏ฟฝ46,656 = b) ๏ฟฝ1,889,5685 =
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) โ๐ง๐ง93
b) โ๐๐6
c) โโ๐๐105
YOU TRY Simplify the following radicals using the exponent rule. Show your work.
a) โ646
b) โ7293
c) ๏ฟฝ๐ฅ๐ฅ2๐ฆ๐ฆ4๐ง๐ง10
d) ๏ฟฝ๐ฅ๐ฅ21๐ฆ๐ฆ427
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 โ8 in a calculator, the calculator would display
2.828427124746190097603377448419โฆ 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 โ no decimals. When we say to simplify an expression with radicals, the simplified expression should have
โข a radical, unless the radical reduces to an integer โข a radicand with no factors containing perfect squares โข no decimals
Following these guidelines ensures the expression is in its simplest form.
Product rule for radicals
If ๐๐,๐๐ are any two positive real numbers, then
โ๐๐๐๐ = โ๐๐ โ โ๐๐ In general, if ๐๐,๐๐ are any two positive real numbers, then
โ๐๐๐๐๐๐ = โ๐๐๐๐ โ โ๐๐๐๐
Where ๐๐ is a positive integer and ๐๐ โฅ ๐๐.
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, ๐๐: โ๐๐๐๐ = ๐๐
For example: โ25 = โ5 โ 5 = โ52 = 5 The product rule for square roots
Given that ๐๐ and ๐๐ are non-negative real numbers, ___________________________________________.
โ45 = ________________________________________________________________________. Example: โ8 = _____________________________________________________________
โ48 = _____________________________________________________________
โ150 = _____________________________________________________________
๏ฟฝ1,350 = _____________________________________________________________
Chapter 12
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MEDIA LESSON Simplify radicals with not perfect radicants โ 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
โ๐๐2๐๐ =
โ๐๐๐๐๐๐๐๐ = When we divide if there is a remainder, the remainder ________________________________________.
Example:
a) โ72 b) โ7503
YOU TRY
Simplify. Show your work.
a) โ75
b) โ2003
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) โ8โ600 b) 3 โโ965
YOU TRY
Simplify. a) 5โ63
b) โ8โ392
Chapter 12
327
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) โ๐๐13๐๐23๐๐10๐๐34
b) ๏ฟฝ125๐ฅ๐ฅ4๐ฆ๐ฆ๐ง๐ง5
YOU TRY
Simplify. Assume all variables are positive. a) ๏ฟฝ๐ฅ๐ฅ6๐ฆ๐ฆ5
b) โ5๏ฟฝ18๐ฅ๐ฅ4๐ฆ๐ฆ6๐ง๐ง10
c) ๏ฟฝ20๐ฅ๐ฅ5๐ฆ๐ฆ9๐ง๐ง6
Chapter 12
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EXERCISE Simplify. Show all your work. Assume all variables are positive.
1) โ245
2) โ36
3) โ12
4) 3โ12
5) 6โ128
6) โ8โ392
7) โ192๐๐
8) โ196๐ฃ๐ฃ2
9) โ252๐ฅ๐ฅ2
10) โโ100๐๐4
11) โ7โ64๐ฅ๐ฅ4
12) โ5โ36๐๐
13) โ4๏ฟฝ175๐๐4
14) 8๏ฟฝ112๐๐2
15) โ2โ128๐๐
16) ๏ฟฝ45๐ฅ๐ฅ2๐ฆ๐ฆ2
17) ๏ฟฝ16๐ฅ๐ฅ3๐ฆ๐ฆ3
18) ๏ฟฝ320๐ฅ๐ฅ4๐ฆ๐ฆ4
19) โ๏ฟฝ32๐ฅ๐ฅ๐ฆ๐ฆ2๐ง๐ง3
20) 5๏ฟฝ245๐ฅ๐ฅ2๐ฆ๐ฆ3
21) โ2โ180๐ข๐ข3๐ฃ๐ฃ
22) โ72๐๐3๐๐4
23) 2๏ฟฝ80โ๐๐4๐๐
24) 6โ50๐๐4๐๐๐๐2
25) 8โ98๐๐๐๐
26) โ512๐๐4๐๐2
27) โ100๐๐4๐๐3
28) โ8๏ฟฝ180๐ฅ๐ฅ4๐ฆ๐ฆ2๐ง๐ง4
29) 2๏ฟฝ72๐ฅ๐ฅ2๐ฆ๐ฆ2
30) โ5๏ฟฝ36๐ฅ๐ฅ3๐ฆ๐ฆ4
Simplify. Show all your work. Assume all variables are positive.
31) โ6253
32) โ7503
33) โ8753
34) โ4 โ964
35) 6 โ1124
36) โ648๐๐24
37) โ224๐๐35
38) ๏ฟฝ224๐๐55
39) โ3 โ896๐๐7
40) โ2 โโ48๐ฃ๐ฃ73
41) โ7 โ320๐๐63
42) ๏ฟฝโ135๐ฅ๐ฅ5๐ฆ๐ฆ33
43) ๏ฟฝโ32๐ฅ๐ฅ4๐ฆ๐ฆ43
44) ๏ฟฝ256๐ฅ๐ฅ4๐ฆ๐ฆ63
45) 7 ๏ฟฝโ81๐ฅ๐ฅ3๐ฆ๐ฆ73
46) 2 โ375๐ข๐ข2๐ฃ๐ฃ83
47) โ3 โ192๐๐๐๐23
48) 6 ๏ฟฝโ54๐๐8๐๐3๐๐73
49) 6 ๏ฟฝ648๐ฅ๐ฅ5๐ฆ๐ฆ7๐ง๐ง24 50) 9๏ฟฝ9๐ฅ๐ฅ2๐ฆ๐ฆ5๐ง๐ง3
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. Letโs look at an example with like terms and like radicals.
2๐ฅ๐ฅ + 5๐ฅ๐ฅ (2 + 5)๐ฅ๐ฅ
7๐ฅ๐ฅ
2โ3 + 5โ3 (2 + 5)โ3
7โ3 Notice that when we combined the terms with โ3, it was similar to combining terms with ๐ฅ๐ฅ. 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
๐๐โ๐๐ ยฑ ๐๐โ๐๐ = (๐๐ ยฑ ๐๐)โ๐๐,
Where ๐๐,๐๐ are real numbers and ๐๐ is some positive real number.
In general, for any root ๐๐, ๐๐โ๐๐๐๐ ยฑ ๐๐โ๐๐๐๐ = (๐๐ ยฑ ๐๐)โ๐๐๐๐ ,
Where ๐๐,๐๐ are real numbers and ๐๐ 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: 2๐ฅ๐ฅ โ 5๐ฆ๐ฆ + 3๐ฅ๐ฅ + 2๐ฆ๐ฆ
_______________________
Simplify: 2โ3 โ 5โ7 + 3โ3 + 2โ7
_______________________
When adding and subtracting radicals, we can ______________________________________________. Example:
a) โ4โ6 + 2โ11 + โ11 โ 5โ6 b) โ53 + 3โ5 โ 8โ53 + 2โ5
YOU TRY
Simplify
a) 7โ65 + 4โ35 โ 9โ35 + โ65
b) โ3โ2 + 3โ5 + 3โ5
Chapter 12
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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 โ2๏ฟฝ50๐ฅ๐ฅ5 + 5๏ฟฝ18๐ฅ๐ฅ5 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) 2โ18 + โ50 b) ๐ฅ๐ฅ ๏ฟฝ๐ฅ๐ฅ2๐ฆ๐ฆ53 + ๐ฆ๐ฆ ๏ฟฝ๐ฅ๐ฅ5๐ฆ๐ฆ23
YOU TRY
Simplify.
a) 5โ45 + 6โ18 โ 2โ98 + โ20
b) 4โ543 โ 9โ163 + 5โ93
Chapter 12
331
EXERCISE Simplify. In this section, we assume all variables to be positive.
1) 2โ5 + 2โ5 + 2โ5
2) โ2โ6 โ 2โ6 โโ6
3) 3โ6 + 3โ5 + 2โ5
4) 2โ2 โ 3โ18 โ โ2
5) 3โ2 + 2โ8 โ 3โ18
6) โ3โ6 โโ12 + 3โ3
7) 3โ18 โ โ2 โ 3โ2
8) โ2โ18โ 3โ8 โ โ20 + 2โ20
9) โ2โ24โ 2โ6 + 2โ6 + 2โ20
10) 3โ24 โ 3โ27 + 2โ6 + 2โ8
11) โ2โ163 + 2โ163 + 2โ23
12) 2โ2434 โ 2โ2434 โ โ34
13) โ6254 -5โ6254 + โ643 โ 5โ643
14) 3โ24 โ 2โ24 โ โ2434
15) โโ3244 + 3โ3244 โ 3โ44
16) 2โ24 + 2โ34 + 3โ644 โ โ34
17) โ3โ65 โ โ645 + 2โ1925 โ 2โ645
18) 2โ1605 โ 2โ1925 โ โ1605 โ โโ1605
19) โโ2566 โ 2โ46 โ 3โ3206 โ 2โ1286
20) 3โ1353 โ โ813 โ โ1353
21) โ3โ18๐ฅ๐ฅ5 โ โ8๐ฅ๐ฅ5 + 2โ8๐ฅ๐ฅ5 + 2โ8๐ฅ๐ฅ5
22) โ2๏ฟฝ2๐ฅ๐ฅ๐ฆ๐ฆ โ ๏ฟฝ2๐ฅ๐ฅ๐ฆ๐ฆ + 3๏ฟฝ8๐ฅ๐ฅ๐ฆ๐ฆ + 3๏ฟฝ8๐ฅ๐ฅ๐ฆ๐ฆ
23) 2โ6๐ฅ๐ฅ2 โ โ54๐ฅ๐ฅ2 โ 3๏ฟฝ27๐ฅ๐ฅ2๐ฆ๐ฆ โ ๏ฟฝ3๐ฅ๐ฅ2๐ฆ๐ฆ
24) 2๐ฅ๐ฅ๏ฟฝ20๐ฆ๐ฆ2 + 7๐ฆ๐ฆโ20๐ฅ๐ฅ2 โ ๏ฟฝ3๐ฅ๐ฅ๐ฆ๐ฆ
25) 3โ24๐ก๐ก โ 3โ54๐ก๐ก โ 2โ96๐ก๐ก + 2โ150๐ก๐ก
Chapter 12
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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 ๐๐,๐๐ are any two positive real numbers, then
โ๐๐๐๐ = โ๐๐ โ โ๐๐ In general, if ๐๐,๐๐ are any two positive real numbers, then
โ๐๐๐๐๐๐ = โ๐๐๐๐ โ โ๐๐๐๐
Where ๐๐ is a positive integer and ๐๐ โฅ ๐๐.
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): pโ๐ข๐ข๐๐ โ ๐๐ โ๐ฃ๐ฃ๐๐ = ________________
Example 1: โ2 โ โ3 = ______________________________________
Example 2: 3โ53 โ 4โ73 = ____________________________________
Multiply:
a) โ15 โ โ6 b) โ183 โ โ603
c) 3โ12 โ 5โ63
d) โ2โ404 โ 7โ184
e) โโ6 ยท โ3โ6
Chapter 12
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YOU TRY
Simplify: a) โ5โ14 โ 4โ6
b) 2 โ183 โ 6 โ153
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) โ18๐ฅ๐ฅ3 โ โ30๐ฅ๐ฅ2 b) โ16๐ฅ๐ฅ23 โ โ81๐ฅ๐ฅ23
YOU TRY
Simplify.
a) โ8๐ฅ๐ฅ25 โ โ4๐ฅ๐ฅ35
b) โ60๐ฅ๐ฅ4 โ โ6๐ฅ๐ฅ7
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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: โ7๏ฟฝโ14 โ โ2๏ฟฝ โ3๏ฟฝ5 + โ3๏ฟฝ
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) โ๐ฅ๐ฅ๏ฟฝ2โ๐ฅ๐ฅ โ 3๏ฟฝ
b) 4๏ฟฝ๐ฆ๐ฆ๏ฟฝ5๏ฟฝ๐ฅ๐ฅ๐ฆ๐ฆ3 โ ๏ฟฝ๐ฆ๐ฆ3๏ฟฝ
c) โ๐ง๐ง3 ๏ฟฝโ๐ง๐ง23 โ 7โ๐ง๐ง53 + 2 โ๐ง๐ง83 ๏ฟฝ
YOU TRY
Simplify.
a) 7โ6 (3โ10 โ 5โ15)
b) โ3๏ฟฝ7โ15๐ฅ๐ฅ3 + 8๐ฅ๐ฅโ60๐ฅ๐ฅ๏ฟฝ
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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: (๐๐ + ๐๐)(๐๐ + ๐๐) = ____________________________________
Always be sure your final answer is ____________________________.
Example: a) ๏ฟฝ3โ7 โ 2โ5๏ฟฝ๏ฟฝโ7 + 6โ5๏ฟฝ
b) ๏ฟฝ2 โ93 + 5๏ฟฝ ๏ฟฝ4 โ33 โ 1๏ฟฝ
MEDIA LESSON Multiply binomials with radicals with variables (Duration 5:29)
View the video lesson, take notes and complete the problems below
Example: a) ๏ฟฝ2โ๐ฅ๐ฅ + 3๏ฟฝ๏ฟฝ5โ๐ฅ๐ฅ โ 4๏ฟฝ b) ๏ฟฝ3๐ฅ๐ฅ2 + โ๐ฅ๐ฅ23 ๏ฟฝ ๏ฟฝ2 โ๐ฅ๐ฅ3 โ 1๏ฟฝ
YOU TRY
Simplify.
a) (โ5 โ 2โ3)(4โ10 + 6โ6)
b) ๏ฟฝ3โ๐ฃ๐ฃ + 2โ3๏ฟฝ๏ฟฝ5โ๐ฃ๐ฃ โ 7โ3๏ฟฝ
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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: (๐๐ + ๐๐)2 = ________________________________
Always be sure your final answer is _________________________
Example:
a) ๏ฟฝโ6 โโ2๏ฟฝ2
b) ๏ฟฝ2 + 3โ7๏ฟฝ2
Conjugates
Recall the Difference of for two squares formula: (๐๐ โ ๐๐)(๐๐ + ๐๐) = ๐๐๐๐ โ ๐๐๐๐ Notice in the 2 factors (๐๐ โ ๐๐) and (๐๐ + ๐๐) 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 โ 5 3 + 5 ๐ฅ๐ฅ + 5 ๐ฅ๐ฅ โ 5
1 โ โ2 1 + โ2 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: (๐๐ โ ๐๐)(๐๐ + ๐๐) = ____________________________________
๏ฟฝ3 โ โ6๏ฟฝ๏ฟฝ3 + โ6๏ฟฝ = ____________________________________________________________________
๏ฟฝโ2 โโ5๏ฟฝ๏ฟฝโ2 + โ5๏ฟฝ = _________________________________________________________________
๏ฟฝ2โ3 + 3โ7๏ฟฝ๏ฟฝ2โ3 โ 3โ7๏ฟฝ = ____________________________________________________________
= ____________________________________________________________
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YOU TRY
a) Simplify: (5โ7 + โ2)2
b) Simplify: (8 โโ5)(8 + โ5)
E. SIMPLIFY QUOTIENTS WITH RADICALS
Quotient rule for radicals
If ๐๐,๐๐ are any two positive real numbers, where ๐๐ โ ๐๐, then
๏ฟฝ๐๐๐๐
=โ๐๐โ๐๐
If ๐๐,๐๐ are any two positive real numbers, where ๐๐ โ ๐๐, then
๏ฟฝ๐๐๐๐
๐๐=โ๐๐๐๐
โ๐๐๐๐
Where ๐๐ is a positive integer and ๐๐ โฅ ๐๐.
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) ๏ฟฝ7516
b) ๏ฟฝ3244
3
Chapter 12
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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) ๏ฟฝ100๐ฅ๐ฅ5๐ฅ๐ฅ
, assume ๐ฅ๐ฅ is positive
b) ๏ฟฝ64๐ฅ๐ฅ2๐ฆ๐ฆ53
๏ฟฝ4๐ฆ๐ฆ23 , assume ๐ฆ๐ฆ 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 + โ175
10
b) 8 โ โ48
6
YOU TRY
Simplify.
a) โ3+โ27
3
b) ๏ฟฝ44๐ฆ๐ฆ6๐๐4
๏ฟฝ9๐ฆ๐ฆ2๐๐8
c) 15 โ1083
20 โ23
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EXERCISE Simplify. Assume all variables are positive.
1) โ4โ16 โ 3โ5
2) 3โ10 โ โ20
3) โ5โ10๐๐2 โ โ5๐๐3
4) โ12๐๐ โ โ15๐๐
5) 3โ4๐๐43 โ โ10๐๐33
6) โ4๐ฅ๐ฅ33 โ โ2๐ฅ๐ฅ43
7) โ6(โ2 + 2)
8) 5โ10(5๐๐ + โ2)
9) โ5โ15(3โ3 + 2)
10) 5โ15(3โ3 + 2)
11) โ10(โ5 + โ2)
12) โ15(โ5 โ 3โ3๐ฃ๐ฃ)
13) (2 + 2๏ฟฝ2)(โ3 + โ2)
14) (โ2 + โ3)(โ5 + 2โ3)
15) (โ5 โ 4โ3)(โ3โ 4โ3)
16) (โ5 โ 5)(2โ5 โ 1)
17) (โ2๐๐ + 2โ3๐๐)(3โ2๐๐ + โ5๐๐)
18) (5โ2 โ 1)(โโ2๐๐ + 5)
19) โ10โ6
20) โ5
4โ125
21) โ125โ100
22) โ53
4 โ43
23) 2โ43โ3
24) 3 โ103
5 โ273
25) ๏ฟฝ12๐๐2
๏ฟฝ3๐๐
26) 4+ 8โ452โ4
27) 3+ โ12โ3
28) 4โ2โ23โ32
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29) 4โโ30โ15
30) 5 โ5๐๐44
โ8๐๐24
31) 5๐ฅ๐ฅ2
4๐ฅ๐ฅ๐ฆ๐ฆ๏ฟฝ3๐ฅ๐ฅ๐ฆ๐ฆ 32) (5 + 2โ6)2
33) (๐ฅ๐ฅ โ ๐ฅ๐ฅโ5)2 34) (โ3 โ โ7)2
35) (5โ6 + 2โ3)2
36) (โ2 โ โ5)(โ2 + โ5)
37) (โ๐ฅ๐ฅ โ ๏ฟฝ๐ฆ๐ฆ)(โ๐ฅ๐ฅ + ๏ฟฝ๐ฆ๐ฆ)
38) (4 โ 2โ3)(4 + 2โ3)
39) (๐ฅ๐ฅ โ ๐ฆ๐ฆโ3)(๐ฅ๐ฅ + ๐ฆ๐ฆโ3)
40) (9โ๐ฅ๐ฅ + ๏ฟฝ๐ฆ๐ฆ)(9โ๐ฅ๐ฅ โ ๏ฟฝ๐ฆ๐ฆ)
Chapter 12
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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.,
๐๐โ๐๐
โโ๐๐โ๐๐
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
โ10 b) 35
3โ7
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) โ7๐๐๐๐โ6๐๐๐๐2
b) ๏ฟฝ 5๐ฅ๐ฅ๐ฆ๐ฆ3
15๐ฅ๐ฅ๐ฆ๐ฆ๐ฅ๐ฅ
YOU TRY
Simplify.
a) โ6โ5
b) 6โ1412โ22
c) โ3โ92โ6
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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 โ Monomial higher root
Use the ____________________
To clear radicals _____________ by extra needed factors in denominator (multiply by the same on top!)
Hint: ___________________ numbers!
Example:
a) 5โ๐๐27
b) ๏ฟฝ 79๐๐2๐๐
3
YOU TRY
Simplify.
a) 4 โ23
7 โ253
b) 3 โ114
โ24
Chapter 12
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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:
(๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐2 โ ๐๐2
Rationalize denominators using the conjugate
We rationalize denominators of the type ๐๐ ยฑ โ๐๐ by multiplying the numerator and denominator by their conjugates, e.g.,
1๐๐ + โ๐๐
โ๐๐ โ โ๐๐๐๐ โ โ๐๐
=๐๐ โ โ๐๐
(๐๐)2 โ (โ๐๐)2
The conjugate for โข ๐๐ + โ๐๐ is ๐๐ โ โ๐๐ โข ๐๐ โ โ๐๐ is ๐๐ + โ๐๐
The case is similar for when there is something like โ๐๐ ยฑ โ๐๐ 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 โ Binomials
What doesnโt work: 1
2+โ3
Recall: ๏ฟฝ2 + โ3๏ฟฝ _______________________
Multiply by the ________________________
Example:
a) 6
5โโ3 b)
3โ5โ24+2โ2
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) โ2
4+โ10
YOU TRY
Simplify.
a) 2
โ3โ5
b) 3โโ52โโ3
c) 2โ5โ3โ75โ6+4โ2
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EXERCISE Simplify. Assume all variables are positive.
1) 2โ43โ3
2) โ12โ3
3) โ23โ5
4) 4โ3โ15
5) 4+2โ3โ9
6) โ53
4 โ43
7) 2โ23 8)
6 โ23
โ93 9) 8
โ3๐ฅ๐ฅ23
10) 2๐ฅ๐ฅโ๐ฅ๐ฅ3 11)
๐ฃ๐ฃโ2๐ฃ๐ฃ34 12)
1โ5๐ฅ๐ฅ4
13) 4+2โ35โ4
14) 2โ5โ54โ13
15) โ2โ3โ3
โ3
16) 5
3โ5+โ2 17)
25+โ2
18) 3
4โ3โ3
19) 4
3+โ5 20) โ 4
4โ4โ2 21)
45 + โ5๐ฅ๐ฅ2
22) 5
2+โ5๐๐3 23)
2โโ5โ3+โ5
24) โ3+โ22โ3โโ2
25) 4โ2+33โ2+โ3
26) 5
โ3+4โ5 27)
2โ5+โ31โโ3
28) ๐๐โ๐๐
โ๐๐โโ๐๐ 29)
7โ๐๐+โ๐๐
30) ๐๐โโ๐๐๐๐+โ๐๐
Chapter 12
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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 ๐๐ โฅ ๐๐ and ๐๐ โฅ ๐๐, then
โ๐๐ = ๐๐ if and only if ๐๐ = ๐๐๐๐ If ๐๐ โฅ ๐๐ and ๐๐ is a real number, then
โ๐๐๐๐ = ๐๐ if and only if ๐๐ = ๐๐๐๐ 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.
๏ฟฝโ๐ฅ๐ฅ๏ฟฝ2
= ___________ ๏ฟฝโ๐ฅ๐ฅ3 ๏ฟฝ3
= ____________
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Example: Solve. a) โ๐ฅ๐ฅ โ 7 = 11
b) โ3๐ฅ๐ฅ + 2 โ 7 = 0
c) 2โ5๐ฅ๐ฅ โ 13 โ 8 = 0
d) โ๐ฅ๐ฅ + 6 = ๐ฅ๐ฅ
YOU TRY
Solve for ๐ฅ๐ฅ.
a) โ7๐ฅ๐ฅ + 2 = 4
b) โ๐ฅ๐ฅ + 3 = 5
c) ๐ฅ๐ฅ + โ4๐ฅ๐ฅ + 1 = 5
d) โ๐ฅ๐ฅ + 6 = ๐ฅ๐ฅ + 4
Chapter 12
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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) โ2๐ฅ๐ฅ + 3 โ โ๐ฅ๐ฅ โ 8 = 0 b) 3 + โ๐ฅ๐ฅ โ 6 = โ๐ฅ๐ฅ + 9
Chapter 12
349
MEDIA LESSON Solve equations with two radicals โ part 2 (Duration 4:33 )
View the video lesson, take notes and complete the problems below
Example: Solve the equation. โ1 โ 8๐ฅ๐ฅ โ โโ16๐ฅ๐ฅ โ 12 = 1
MEDIA LESSON Solve equations with two radicals โ part 3 โ 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 ๐ฅ๐ฅ and check solutions
a) โ2๐ฅ๐ฅ + 1 โ โ๐ฅ๐ฅ = 1
Check solutions
b) โ2๐ฅ๐ฅ + 6 โ โ๐ฅ๐ฅ + 4 = 1
Check solutions
Chapter 12
351
C. RADICAL EQUATIONS WITH HIGHER ROOTS
MEDIA LESSON Solve equations with radicals โ 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 ______________________________.
โ๐ฅ๐ฅ3 = 4 then ๐ฅ๐ฅ =_______
Example: a) โ2๐ฅ๐ฅ โ 53 = 6 b) โ4๐ฅ๐ฅ โ 75 = 2
YOU TRY
Solve for ๐๐.
a) โ๐๐ โ 13 = โ4
b) โ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ4 = 2
Chapter 12
352
EXERCISE Solve. Be sure to verify all solutions.
1) โ2๐ฅ๐ฅ + 3 โ 3 = 0
2) โ6๐ฅ๐ฅ โ 5 โ ๐ฅ๐ฅ = 0
3) 3 + ๐ฅ๐ฅ = โ6๐ฅ๐ฅ + 13
4) โ3 โ 3๐ฅ๐ฅ โ 1 = 2๐ฅ๐ฅ
5) โ4๐ฅ๐ฅ + 5 โ โ๐ฅ๐ฅ + 4 = 2
6) โ2๐ฅ๐ฅ + 4 โ โ๐ฅ๐ฅ + 3 = 1
7) โ2๐ฅ๐ฅ + 6 โ โ๐ฅ๐ฅ + 4 = 1
8) โ6 โ 2๐ฅ๐ฅ โ โ2๐ฅ๐ฅ + 3 = 3
9) โ5๐ฅ๐ฅ + 1 โ 4 = 0 10) โ๐ฅ๐ฅ + 1 = โ๐ฅ๐ฅ + 1
11) ๐ฅ๐ฅ โ 1 = โ7 โ ๐ฅ๐ฅ
12) โ2๐ฅ๐ฅ + 2 = 3 + โ2๐ฅ๐ฅ โ 1
13) โ3๐ฅ๐ฅ + 4 โ โ๐ฅ๐ฅ + 2 = 2
14) โ7๐ฅ๐ฅ + 2 โ โ3๐ฅ๐ฅ + 6 = 6
15) โ4๐ฅ๐ฅ โ 3 = โ3๐ฅ๐ฅ + 1 + 1
16) โ๐ฅ๐ฅ + 2 โ โ๐ฅ๐ฅ = 2
17) โ๐ฅ๐ฅ + 25 = โโ35
18) โ5๐ฅ๐ฅ + 13 โ 2 = 4
19) 3โ๐ฅ๐ฅ3 = 12
20) โ7๐ฅ๐ฅ + 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
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