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CHAPTER 1 2 : 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
- CHAPTER 12: RADICALS Contents
- SECTION 12.1 INTRODUCTION TO RADICALS
- A. INTRODUCTION TO PERFECT SQUARES AND PRINCIPAL SQUARE ROOT
- B. INTRODUCTION TO RADICALS
- C. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏𝒏𝒏𝒏𝒏 ROOT
- D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏𝒏𝒏𝒏𝒏 ROOT USING EXPONENT RULE
- E. SIMPLIFY RADICALS WITH NO PERFECT ROOT
- F. SIMPLIFY RADICALS WITH COEFFICIENTS
- G. SIMPLIFY RADICALS WITH VARIABLES WITH NO PERFECT RADICANTS
- EXERCISE
- SECTION 12.2: ADD AND SUBTRACT RADICALS
- A. ADD AND SUBTRACT LIKE RADICALS
- B. SIMPLIFY, THEN ADD AND SUBTRACT LIKE RADICALS
- EXERCISE
- SECTION 12.3: MULTIPLY AND DIVIDE RADICALS
- A. MULTIPLY RADICALS WITH MONOMIALS
- B. DISTRIBUTE WITH RADICALS
- C. MULTIPLY RADICALS USING FOIL
- D. MULTIPLY RADICALS WITH SPECIAL-PRODUCT FORMULAS
- E. SIMPLIFY QUOTIENTS WITH RADICALS
- EXERCISE
- SECTION 12.4: RATIONALIZE DENOMINATORS
- A. RATIONALIZING DENOMINATORS WITH SQUARE ROOTS
- B. RATIONALIZING DENOMINATORS WITH HIGHER ROOTS
- C. RATIONALIZE DENOMINATORS USING THE CONJUGATE
- EXERCISE
- SECTION 12.5: RADICAL EQUATIONS
- A. RADICAL EQUATIONS WITH SQUARE ROOTS
- B. RADICAL EQUATIONS WITH TWO SQUARE ROOTS
- C. RADICAL EQUATIONS WITH HIGHER ROOTS
- EXERCISE
- CHAPTER REVIEW
YOU TRY
a) Find the perfect square of:
2
= ________________
2
= ________________
2
= ________________
2
= ________________
2
= ________________
2
= ________________
2
= ________________
2
= ________________
2
=________________
2
=________________
b) Find the square root of:
√ 441 = _______________
√484 =_______________
√529 =_______________
√576 =_______________
√625 =_______________
√676 =_______________
√729 =_______________
√784 =_______________
√841 = _______________
√ 900 = _______________
MEDIA LESSON
Principal n
th
square roots vs. general square roots (Duration 5:23 )
Note: In this class, we will only consider the principal 𝒏𝒏
𝒏𝒏𝒏𝒏
roots 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 = 𝒃𝒃
𝟐𝟐
The √ is the radical symbol, and 𝒂𝒂 is called the radicand.
If given something like √𝒂𝒂
𝟑𝟑
, then 3 is called the root or index ; hence, √𝒂𝒂
𝟑𝟑
is called the cube root or
third root of 𝒂𝒂. In general,
√𝒂𝒂
𝒏𝒏
= 𝒃𝒃 if and only if 𝒂𝒂 = 𝒃𝒃
𝒏𝒏
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:
√ 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.
MEDIA LESSON
Introduction to square roots, cube roots, and N
th
roots (Duration 9:09)
View the video lesson, take notes and complete the problems below
The principal 𝒏𝒏
𝒏𝒏𝒏𝒏
root of 𝒂𝒂 is the 𝒏𝒏
𝒏𝒏𝒏𝒏
root that has the same sign as 𝒂𝒂, and it is denoted by the radical
symbol.
𝒏𝒏
We read this as the “___________________________”, “______________”, or “_______________”.
The positive integer ______________________________ of the radical. If 𝑛𝑛 = 2, ____________ the index.
The number _______________________.
√ 4 =________________
=_______________
−√ 4 =________________
=_______________
Square roots (n = 2)
√ 1 =________________________________ −√ 1 =________________________________
4 = ________________________________ −
4 = ________________________________
9 = ________________________________ −
9 = ________________________________
√ 16 = _______________________________ −√ 16 = _______________________________
√ 25 = _______________________________ −√ 25 = _______________________________
Cube roots (n = 3)
3
= __________________________ √− 1
3
= __________________________
3
= __________________________
3
= __________________________
3
=__________________________
3
=_________________________
3
= __________________________ √− 64
3
= _________________________
3
=_________________________ √− 125
3
=________________________
Example: Simplify
3
MEDIA LESSON
Simplify perfect 𝒏𝒏
𝒏𝒏𝒏𝒏
roots – negative radicands (Duration 4:32 )
View the video lesson, take notes and complete the problems below
Example: Simplify each of the following.
a) √ 16
= ________________________________________________________________________
b) √− 32
5
= ________________________________________________________________________
c) √− 64
= ________________________________________________________________________
YOU TRY
Simplify. Show your work.
a) √
b) √
3
c) − √
d) √ 1
5
D. SIMPLIFY RADICALS WITH PERFECT PRINCIPAL 𝒏𝒏
𝒏𝒏𝒏𝒏
ROOT USING EXPONENT RULE
There is a more efficient way to find the 𝑛𝑛
𝑡𝑡ℎ
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
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: √𝒎𝒎
where 𝒏𝒏 is the _______________
Roots of an expression with exponents: _________________the ________________ by the __________.
Example: Simplify.
a) � 46 , 656 =
b) � 1 , 889 , 568
5
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) √𝑧𝑧
9
b) √𝑚𝑚
6
c) − √𝑛𝑛
10
5
YOU TRY
Simplify the following radicals using the exponent rule. Show your work.
a)
6
b) √ 729
3
c) �𝑥𝑥
2
4
10
d) �𝑥𝑥
21
𝑦𝑦
42
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
2
𝑛𝑛
𝑛𝑛
When we divide if there is a remainder, the remainder ________________________________________.
Example:
a) √
b) √ 750
3
YOU TRY
Simplify. Show your work.
a) √
b) √
3
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 √
5
YOU TRY
Simplify.
a) 5 √ 63 b) − 8 √ 392
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) √𝑎𝑎
13
23
10
3
4
b) � 125 𝑥𝑥
4
5
YOU TRY
Simplify. Assume all variables are positive.
a) �𝑥𝑥
6
5
b) − 5 � 18 𝑥𝑥
4
6
10
c) � 20 𝑥𝑥
5
9
6
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.
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
Where 𝒂𝒂, 𝒃𝒃 are real numbers and 𝒙𝒙 is some positive real number.
In general, for any root 𝒏𝒏,
𝒏𝒏
𝒏𝒏
𝒏𝒏
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: 2 𝑥𝑥 − 5 𝑦𝑦 + 3 𝑥𝑥 + 2 𝑦𝑦
_______________________
Simplify: 2 √
_______________________
When adding and subtracting radicals, we can ______________________________________________.
Example:
a) − 4 √ 6 + 2 √ 11 + √ 11 − 5 √ 6 b) √ 5
3
3
YOU TRY
Simplify
a) 7 √ 6
5
5
5
5
b) − 3 √ 2 + 3 √ 5 + 3 √ 5
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
5
5
/\
/\
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 �𝑥𝑥
2
5
3
5
2
3
YOU TRY
Simplify.
a) 5 √ 45 + 6 √ 18 − 2 √ 98 + √ 20 b) 4 √ 54
3
3
3
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
√𝑎𝑎𝑎𝑎
𝑛𝑛
=
√𝑎𝑎
𝑛𝑛
∙ √𝑎𝑎
𝑛𝑛
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 √𝑢𝑢
𝑛𝑛
𝑛𝑛
= ________________
Example 1: √ 2 ⋅ √ 3 = ______________________________________
Example 2: 3 √ 5
3
3
= ____________________________________
Multiply:
a) √ 15 ⋅ √ 6 b) √ 18
3
3
c) 3 √ 12 ⋅ 5 √ 63 d) − 2 √ 40
4
4
e) − √
YOU TRY
Simplify:
a) − 5 √
b) 2 √ 18
3
3
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) √ 18 𝑥𝑥
3
2
b) √ 16 𝑥𝑥
2
3
2
3
YOU TRY
Simplify.
a) √ 8 𝑥𝑥
2
5
3
5
b) √ 60 𝑥𝑥
4
7
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 √ 9
3
3
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 √
b) � 3 𝑥𝑥
2
2
3
3
YOU TRY
Simplify.
a) (√ 5 − 2 √ 3 )( 4 √ 10 + 6 √ 6 ) b) � 3 √
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) � √
2
b) � 2 + 3 √
2
Conjugates
Recall the Difference of for two squares formula:
𝟐𝟐
𝟐𝟐
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
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:
= ____________________________________
� 3 − √ 6 ��3 + √ 6 � = ____________________________________________________________________
�√ 2 − √ 5 ��√2 + √ 5 � = _________________________________________________________________
� 2 √3 + 3√ 7 �� 2 √ 3 − 3 √ 7 � = ____________________________________________________________
= ____________________________________________________________
