1

Lesson 2.1: Simplifying, Adding and Subtracting Radicals

Learning Goals:

1. How do we simplify radical expressions? 2. How do we add and subtract radical expressions?

Are the following numbers rational, irrational, or neither one?

1. −√25 rational 2. √35 irrational

3. √−8 neither 4. √−83

rational

Rational Numbers:

numbers that can be expressed in the form 𝑎

𝑏 where 𝑏 ≠ 0

(i.e., fraction, integers, terminating decimals, repeating decimals)

Irrational Numbers:

numbers that cannot be expressed as a fraction.

index = 2 index = 3

√4 = 2 √83

= 2

2 ∙ 2 2 ∙ 2 ∙ 2

2

√64 vs. √643

to get the “3” use MATH #4 8 vs. 4

Directions: Express in simplest radical form.

1. √32 2. 5√40

index = 2 index = 2

√16 ∙ 2 5 ∙ √4 ∙ 10

4 ∙ √2 5 ∙ 2 ∙ √10

10√10

What is the difference between these two questions? Different index!

3) √52𝑥6𝑏5 index = 2 √52𝑥6𝑏53 index = 3

√4 ∙ 13 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 √2 ∙ 2 ∙ 13 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏3

(groups of two) (groups of three)

2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ √13 ∙ 𝑏 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ √2 ∙ 2 ∙ 13 ∙ 𝑏 ∙ 𝑏3

2𝑥3𝑏2√13𝑏 𝑥2𝑏 √52𝑏23

4. Based on the previous question, explain how you would solve the following

problem? Then simplify the radical expression.

√32𝑎3𝑏44

√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏4

2 ∙ 𝑏 ∙ √2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎4

2𝑏 √2𝑎34

3

How can solving the first question using the power law, help us to solve the

second question? Distribute the radical!

5. (5𝑏

12)

2 √

4𝑎2

9

(51𝑏1

121 )2

√4𝑎2

√9

52𝑏2

122 2𝑎

3

25𝑏2

144

6. Based on the previous question, explain how you would solve the following

problem? Then simplify the radical expression.

√8

125𝑦6

3

=2

5𝑦2

Polynomials cannot use the power law!

7. √30𝑥3 + 34𝑥3 8. √9𝑎2 + 9𝑏2

√64𝑥3 combine like terms! √9(𝑎2 + 𝑏2) GCF = 9

√8 ∙ 8 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 √9 ∙ √𝑎2 + 𝑏2

8𝑥√𝑥 3√𝑎2 + 𝑏2

4

Important information to remember:

Radicals that are simplified have:

No fractions left under the radical symbol.

No perfect power factors in the radicand, 𝑘.

No exponents in the radicand, 𝑘, greater than the

index, 𝑛.

No radicals appearing in the denominator of a

fractional answer.

Adding and Subtracting Radicals

1. 4𝑥 − 𝑥 2. 3𝑥 + 2𝑥 − 9𝑥 3. 7𝑥 + 10𝑦 − 3𝑥

3𝑥 −4𝑥 10𝑦 + 4𝑥

What was your method for simplifying # 1 – 3?

Combine like terms! Same variable/exponent

How do you think it relates to our lesson today on adding and subtracting

radicals?

Combine like radicals (terms) – Same index and same radicand.

To add or subtract radicals:

You must have the same index and radicand! Expand/Reduce radicals 1st.

Directions: Add or subtract the following, express all answers in simplest radical

form.

1. 5√11𝑥3

− √11𝑥3

2. 2√300 + 4√75

4√11𝑥3

cannot add until radicands are the same

2√100 ∙ 3 + 4√25 ∙ 3

2 ∙ 10 ∙ √3 + 4 ∙ 5 ∙ √3

20√3 + 20√3

40√3

5

3. √160𝑎44+ √10𝑎44

4. 𝑎√45 + √20𝑎2 − 5√2𝑎

√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎4 + √2 ∙ 5 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎4 𝑎√3 ∙ 3 ∙ 5 + √2 ∙ 2 ∙ 5 ∙ 𝑎 ∙ 𝑎 − 5√2𝑎

2𝑎 √2 ∙ 54

+ 𝑎 √2 ∙ 54

3𝑎√5 + 2𝑎√5 − 5√2𝑎

2𝑎 √104

+ 𝑎 √104

5𝑎√5 − 5√2𝑎

3𝑎 √104

5. √1

4+ √

9

4− √45 6. √9𝑥3 + √27𝑥2

√1

√4+

√9

√4− √9 ∙ 5 √3 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 + √3 ∙ 3 ∙ 3 ∙ 𝑥 ∙ 𝑥

1

2+

3

2− 3√5 3𝑥√𝑥 + 3𝑥√3

4

2− 3√5

2 − 3√5

6

Homework 2.1: Simplifying Radical Expressions

Determine whether the following are rational, irrational, or neither:

1. √8 2. −√16 3. √−16

Write each of the following in simplest radical form:

4. √80𝑐54 5. 5√200𝑥𝑦2 6. √

7

27𝑦6

3

Add or subtract each of the following and place answers in simplest radical form.

7. 3√5𝑦 − √20𝑦 8. √12 − √243

+ √48 + √813

9. √1

4+ √

25

4− √90

7

Lesson 2.2: Multiplying and Dividing Radical Expressions

Learning Goals:

1) How do we add and subtract radical expressions?

2) How do we multiply and divide radical expressions?

Warm-Up:

Using your calculator, find the values of each of the following to two decimal

places.

a. √3 ∙ √5 = 3.87 b. √15 = 3.87

c. 2√2 ∙ 5√3 = 24.49 d. 10√6 = 24.49

e. 4√6 ∙ 2√6 = 39.19 f. 8√36 = 39.19

g. √10

√2 = 2.24 h. √5 = 2.24

i. 25√21

5√3 = 13.23 j. 5√7 = 13.23

Based on the examples above, finish the rules below for multiplying/dividing

radical expressions: when you multiply/divide, you do not need like radicals!

√𝑎𝑛

∙ √𝑏𝑛

= √𝑎 ∙ 𝑏𝑛

√𝑎

𝑛

√𝑏𝑛 = √

𝑎

𝑏

𝑛

√3 ∙ √5 = √15 √10

√2= √5

8

Multiplying/Dividing Racial Expressions

Express each of the following in simplest radical form:

1. √213

∙ √4

3

3 Multiply 1st! 2.

√324𝑎4𝑏33

√6𝑎𝑏23 Divide 1st!

√21 ∙4

3

3 √

324𝑎4𝑏3

6𝑎𝑏2

3

√283

(reduce?) √54𝑎3𝑏3

(reduce?)

√7 ∙ 2 ∙ 23

(can’t) √3 ∙ 3 ∙ 3 ∙ 2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏3

3𝑎√2𝑏3

3. 4√2+8√12

2√2 separate into 2 fractions 4. 2𝑥√3𝑥2𝑦 ∙ 3√15𝑥𝑦3

4√2

2√2+

8√12

2√2 (2𝑥 ∙ 3)√(3𝑥2𝑦 ∙ 15𝑥𝑦3)

2 + 4√6 unlike terms! 6𝑥√3 ∙ 3 ∙ 5 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦

18𝑥2𝑦2√5𝑥

5. (3√2 + √7)(3√2 − √7) = OR by a 2 𝑥 2 Table

9√4−3√14 + 3√14 − √49 =

9(2) − 7 =

18 − 7 =

11

6. Express 2√6(8 − 4√3) as a product in simplest form.

16√6 − 8√18 = 16√6 − 8√9 ∙ 2 = 16√6 − 8(3)√2 = 16√6 − 24√2

9

Homework 2.2: Multiplying and Dividing Radical Expressions

1. (3 + √2)2 2. √𝑏

5∙ √

𝑏7

5

3. (2 + 2√5)(2 − 2√5) 4. 6√3+10√21

2√3

5. √432𝑎7𝑏53

√8𝑎2𝑏33 6. 2−√5

7+√5 [use conjugates]

7. Express 4√5(−2√8 + 6√5) as a product in simplest form.

8. 3𝑥3√80 + 2√125𝑥6 9. √5

8

10

Lesson 2.3: Rationalizing a Denominator

Learning Goal: What do you think it means to rationalize the denominator of a

fraction? Get rid of the radicals in the denominator!

Conjugate is a binomial formed by negating the second sign of the binomial

Example: 𝑥 − 3 and 𝑥 + 3 or 𝑥 + √5 and 𝑥 − √5

(𝑥 − 3)(𝑥 + 3) = 𝑥2 + 3𝑥 − 3𝑥 − 9 = 𝑥2 − 9

(𝑥 + √5)(𝑥 − √5) = 𝑥2−𝑥√5 + 𝑥√5 − √25 = 𝑥2 − 5

Rationalizing the denominator:

1. 4

2√3=

4∙√3

2√3∙√3=

4√3

6=

2√3

3 2. √

𝑎3

2= √

𝑎3∙2

2∙2=

𝑎√2𝑎

2

Use the conjugate!

3. √3

5−2√3=

√3

5−2√3∙

5+2√3

5+2√3=

5√3+2√9

25+10√9−10√9−4√9=

5√3+2(3)

25−4(3)=

6+5√3

13

4. 2+√2

3+√2=

2+√2

3+√2∙

3−√2

3−√2=

6−2√2+3√2−√4

9−3√2+3√2−√4=

6+√2−2

9−2=

4+√2

7

5. Simplify: 3−√3

3+√3 =

3−√3

3+√3∙

3−√3

3−√3=

9−6√3+3

9−3=

12−6√3

6= 2 − √3

6. The expression 3+4√5

4−2√5 is equivalent to

3 + 4√5

4 − 2√5∙

4 + 2√5

4 + 2√5=

12 + 6√5 + 16√5 + 8(5)

16 − 4(5)=

42 + 22√5

−4=

21 + 11√5

−2

11

Homework 2.3: Rationalizing a Denominator

1. The expression 3−√8

√3 is equivalent to

(1) √3−2√6

√3 (2) −√3 +

2

3√6 (3)

3−√24

3 (4) √3 −

2

3√6

2. Express 5

3−√2 with a rational denominator, in simplest radical form.

3. Express the reciprocal of 3 − √7 in simplest radical form with a rational

denominator.

4. The expression 7

2+3√2 is equivalent to

(1) −2+3√2

2 (2)

2−3√2

2 (3) −2 + 3√2 (4) 2 − 3√2

5. Expressed in simplest form, 2√3

1−√3is equivalent to

(1) −3 − √3 (2) −3 + √3 (3) 2√3 (4) −3

6. The expression 3+5√3

4−2√3 is equivalent to

(1) −9+7√3

2 (2)

21+13√3

2 (3)

−18+14√3

4 (4)

42−26√3

4

12

Lesson 2.4: Rational Exponents

Learning Goals:

1) How do we evaluate radical expressions with rational exponents?

2) How do we rewrite rational exponents as radicals?

Do Now: Simplify the following expressions.

index is 2 index is 3 index is 4 (math #5 on calc)

1. √25 = 5 2. √−83

= −2 3. √814

= 3

4. 251

2 = 5 5. (−8)1

3 = −2 6. 811

4 = 3

Based on the questions above, how can you rewrite the expressions below?

𝑥1

𝑛 = √𝑥𝑛

√𝑥𝑛

= 𝑥1

𝑛

The index of a radical can be written as the denominator of a fractional exponent!

912 = √912

= √9 = 3

Rewrite each of the following using radicals, and then simplify, if possible.

1. 161

4 = √164

= 2 2. 9−1

2 =1

912

=1

√9=

1

3

3. 𝑥−1

5 =1

𝑥15

=1

√𝑥5 4. 4𝑥

1

4 = 4 ∙ 𝑥1

4 = 4√𝑥4

5. (4𝑥)1

2 = 41

2𝑥1

2 = √4 ∗ √𝑥 = 2√𝑥 6. (27

𝑥)

−1

3=

27−

13

𝑥−

13

=𝑥

13

2713

=√𝑥3

√273 =

√𝑥3

3

13

A rational exponent does not have to be in the form 1

𝑛. Other rational numbers,

such as 3

2 or −

4

5 can also be used as exponents.

Denominator = index

Let’s think about the expression 43

2. Evaluate this expression in two different

ways by using the laws of exponents. Then verify these answers with your

calculator.

432 = (√4)

3= 23 = 8

432 = √43 = √64 = 8

432 = 8

Rational Exponent Definition

For any rational number 𝑚

𝑛 we define 𝑏

𝑚

𝑛 to be ( √𝑏𝑛

)𝑚

or √𝑏𝑚𝑛.

Rewrite each of the following using radicals, and then simplify, if possible.

1. 163

4 = (√164

)3

= 23 = 8 2. 253

2 = (√25)3

= 53 = 125

3. 8−2

3 =1

823

=1

( √83

)2 =

1

4

4. (64

125)

−2

3=

64−

23

125−

23

=125

23

6423

=( √125

3)

2

( √643

)2 =

25

16

5. (27𝑝6)5

3 = (√27𝑝63)

5= (3𝑝2)5 = 35𝑝10 = 243𝑝10

14

Practice:

1. Yoshiko said that 161

4 = 4 because 4 is one-fourth of 16. Use properties of

exponents to explain why she is or is not correct.

She is wrong because 161

4 = √164

= 2 and 2 ≠ 4.

2. Rita said that 82

3 = 128 because 82

3 = 82 ∙ 81

3, so 82

3 = 64 ∙ 2, and then

82

3 = 128. Use properties of exponents to explain why she is or is not correct.

She is wrong because 82

3 = (√83

)2

= 22 = 4 and 4 ≠ 128.

3. Simplify the expression (𝑚6)−2

3 and write your answer using a positive

exponent. Use the power law!

(𝑚6)−23 = 𝑚6∙−

23 = 𝑚−4 =

1

𝑚4

4. Simplify the expression below and write your answer using positive exponents

only.

(𝑥

12𝑦−2

𝑦𝑥−74

)

4

=𝑥2𝑦−8

𝑦4𝑥−7=

𝑥2𝑥7

𝑦4𝑦8=

𝑥9

𝑦12

5. Find the expression (𝑥 + 3)1

2 + (𝑥 − 3)0 + (𝑥 + 2)−2

3 when 𝑥 = 6

(6 + 3)12 + (6 − 3)0 + (6 + 2)−

23

912 + 30 + 8−

23

√9 + 1 +1

823

= 3 + 1 +1

(√83

)2 = 4 +

1

4= 4

1

4 or

14

4 or 4.25

15

Homework 2.4: Rational Exponents

1. Find the value of 274

3 2. If 𝑥0 + 𝑥2

3 + 𝑥−2

3, find𝑓(8).

3. If 𝑛 > 0, the expression (1

𝑛)

−2

3is equal to

(1) −𝑛2

3 (2) −𝑛3

2 (3) √𝑛23 (4) √𝑛3

4. Simplify the expression (𝑥8)−3

4 and write your answer using a positive

exponent.

5. Write each expression in the form 𝑏𝑚

𝑛 . If a numeric expression is a rational

number, then write your answer without exponents.

a. 𝑏2

3 ∙ 𝑏1

2 b. (𝑏−1

5)

2

3

c. 641

3 ∙ 643

2 d. (92

42)

3

2

6. Evaluate 𝑥2

3 if 𝑥 = −27

16

Lesson 2.5: Solving Radical Equations

Learning Goals

1) How do we solve radical equations?

2) How do we check for extraneous roots?

When solving a radical equation, the equation derived by squaring both sides is

not equivalent to the given equation. In other words, the derived equation does

not always have the same solution as the original equation. If a number is a root

of the original equation, that number must also be a root of the derived equation,

but the converse of this statement is not true.

It is helpful to demonstrate why an extraneous root occurs.

1. √𝑥 − 2 = 5

(√𝑥 − 2)2

= 52 square both sides to eliminate the radical

𝑥 − 2 = 25 make sure to always check your answer:

𝑥 = 27 √27 − 2 = 5

√25 = 5

5 = 5√

2. √2𝑦 − 1 + 7 = 4

√2𝑦 − 1 = −3 isolate your radical

(√2𝑦 − 1)2

= (−3)2 make sure to always check your answer:

2𝑦 − 1 = 9 √2(5) − 1 + 7 = 4

2𝑦 = 10 √9 + 7 = 4

𝑦 = 5 3 + 7 = 4

10 ≠ 4 therefore it is { } or ∅

17

3. 3√𝑥 − 2 − 2√𝑥 + 8 = 0

3√𝑥 − 2 = 2√𝑥 + 8 make sure to always check your answer:

(3√𝑥 − 2)2

= (2√𝑥 + 8)2 3√10 − 2 − 2√10 + 8 = 0

9(𝑥 − 2) = 4(𝑥 + 8) 3√8 − 2√18 = 0

9𝑥 − 18 = 4𝑥 + 32 3√4 ∙ 2 − 2√9 ∙ 2 = 0

5𝑥 = 50 3 ∙ 2√2 − 2 ∙ 3√2 = 0

𝑥 = 10 6√2 − 6√2 = 0

0 = 0√

4. 𝑥 = 1 + √𝑥 + 5 make sure to always check your answers:

𝑥 − 1 = √𝑥 + 5 isolate the radical 4 = 1 + √4 + 5

(𝑥 − 1)2 = (√𝑥 + 5)2 4 = 1 + √9

(𝑥 − 1)(𝑥 − 1) = 𝑥 + 5 4 = 1 + 3

𝑥2 − 1𝑥 − 1𝑥 + 1 = 𝑥 + 5 4 = 4√

𝑥2 − 2𝑥 + 1 = 𝑥 + 5

𝑥2 − 3𝑥 − 4 = 0 −1 = 1 + √−1 + 5

(𝑥 − 4)(𝑥 + 1) = 0 −1 = 1 + √4

𝑥 − 4 = 0 and 𝑥 + 1 = 0 −1 = 1 + 2

𝑥 = 4 and 𝑥 = −1 −1 ≠ 3

18

Homework 2.5: Solving Radical Equations

1. √5 + 𝑎 = 7 2. 3 − √𝑦 = 1

3. √1 + 𝑥 = 3 4. √3 − 𝑦 = 1

5. . 1 − √𝑥 + 73

= 4 6. 5 + √𝑥 = 2