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Add, Subtract, and Multiply Radical Expressions

8.4 Add, Subtract, and Multiply Radical Expressions

Learning Objectives

By the end of this section, you will be able to:

  • Add and subtract radical expressions
  • Multiply radical expressions
  • Use polynomial multiplication to multiply radical expressions

Be Prepared 8.10

Before you get started, take this readiness quiz.

Add: 3x2+9x−5−(x2−2x+3).3x2+9x−5−(x2−2x+3).
 If you missed this problem, review Example 5.5.

Be Prepared 8.11

Simplify: (2+a)(4−a).(2+a)(4−a).
 If you missed this problem, review Example 5.28.

Be Prepared 8.12

Simplify: (9−5y)2.(9−5y)2.
 If you missed this problem, review Example 5.31.

Add and Subtract Radical Expressions

Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.

Like Radicals

Like radicals are radical expressions with the same index and the same radicand.

We add and subtract like radicals in the same way we add and subtract like terms. We know that 3x+8x3x+8x is 11x.11x. Similarly we add 3x+8x3x+8x and the result is 11x.11x.

Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.

Example 8.36

Simplify: ⓐ 22−7222−72 ⓑ 5y3+4y35y3+4y3 ⓒ 7x4−2y4.7x4−2y4.

Solution

ⓐ

22−7222−72
Since the radicals are like, we subtract the
coefficients.		 −52−52
Example illustrating the subtraction of like radical expressions, with step-by-step explanation.

ⓑ

5y3+4y35y3+4y3
Since the radicals are like, we add the
coefficients.		 9y39y3
This table demonstrates adding like radicals by combining coefficients, showing an initial expression, the rationale, and the simplified result.

ⓒ

7x4−2y47x4−2y4
Table presenting the algebraic expression: 7⁴√x - 2⁴√y.

The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.

Try It 8.71

Simplify: ⓐ 82−9282−92 ⓑ 4x3+7x34x3+7x3 ⓒ 3x4−5y4.3x4−5y4.

Try It 8.72

Simplify: ⓐ 53−9353−93 ⓑ 5y3+3y35y3+3y3 ⓒ 5m4−2m3.5m4−2m3.

For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.

Example 8.37

Simplify: ⓐ 25n−65n+45n25n−65n+45n ⓑ 3xy4+53xy4−43xy4.3xy4+53xy4−43xy4.

Solution

ⓐ

25n−65n+45n25n−65n+45n
Since the radicals are like, we combine them. 05n05n
Simplify. 00
Step-by-step simplification of a radical expression, showing the process of combining like radicals to reach a final value of zero.

ⓑ

3xy4+53xy4−43xy43xy4+53xy4−43xy4
Since the radicals are like, we combine them. 23xy423xy4
Example demonstrating the combination of like radicals.

Try It 8.73

Simplify: ⓐ 7x−77x+47x7x−77x+47x ⓑ 45xy4+25xy4−75xy4.45xy4+25xy4−75xy4.

Try It 8.74

Simplify: ⓐ 43y−73y+23y43y−73y+23y ⓑ 67mn3+7mn3−47mn3.67mn3+7mn3−47mn3.

Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.

Example 8.38

Simplify: ⓐ 20+3520+35 ⓑ 243−3753243−3753 ⓒ 12484−232434.12484−232434.

Solution

ⓐ

20+3520+35
Simplify the radicals, when possible. 4·5+354·5+35
25+3525+35
Combine the like radicals. 5555
Step-by-step simplification of a radical expression involving addition.

ⓑ

243−3753243−3753
Simplify the radicals. 83·33−1253·3383·33−1253·33
233−533233−533
Combine the like radicals. −333−333
Step-by-step simplification of a radical expression involving the subtraction of cube roots.

ⓒ

12484−23243412484−232434
Simplify the radicals. 12164·34−23814·3412164·34−23814·34
12·2·34−23·3·3412·2·34−23·3·34
34−23434−234
Combine the like radicals. −34−34
Step-by-step solution demonstrating the simplification and combination of radical expressions involving fourth roots.

Try It 8.75

Simplify: ⓐ 18+6218+62 ⓑ 6163−225036163−22503 ⓒ 23813−12243.23813−12243.

Try It 8.76

Simplify: ⓐ 27+4327+43 ⓑ 453−7403453−7403 ⓒ 121283−53543.121283−53543.

In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption throughout the rest of this chapter.

Example 8.39

Simplify: ⓐ 950m2−648m2950m2−648m2 ⓑ 54n53−16n53.54n53−16n53.

Solution

ⓐ

950m2−648m2950m2−648m2
Simplify the radicals. 925m2·2−616m2·3925m2·2−616m2·3
9·5m·2−6·4m·39·5m·2−6·4m·3
45m2−24m345m2−24m3
The radicals are not like and so cannot be
combined.
Step-by-step guide on simplifying a mathematical expression involving radical terms, detailing the breakdown of square roots and explaining why the final terms cannot be combined.

ⓑ

54n53−16n5354n53−16n53
Simplify the radicals. 27n33·2n23−8n33·2n2327n33·2n23−8n33·2n23
3n2n23−2n2n233n2n23−2n2n23
Combine the like radicals. n2n23n2n23
Illustrates the step-by-step simplification of a mathematical expression involving the subtraction of two cube roots.

Try It 8.77

Simplify: ⓐ 32m7−50m732m7−50m7 ⓑ 135x73−40x73.135x73−40x73.

Try It 8.78

Simplify: ⓐ 27p3−48p327p3−48p3 ⓑ 256y53−32y53.256y53−32y53.

Multiply Radical Expressions

We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of Roots ‘in reverse’ to multiply square roots. Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Roots so we see both ways together.

Product Property of Roots

For any real numbers, anan and bn,bn, and for any integer n≥2n≥2

abn=an·bnandan·bn=abnabn=an·bnandan·bn=abn

When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply 4x·3y4x·3y we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

Example 8.40

Simplify: ⓐ (62)(310)(62)(310) ⓑ (−543)(−463).(−543)(−463).

Solution

ⓐ

(62)(310)(62)(310)
Multiply using the Product Property. 18201820
Simplify the radical. 184·5184·5
Simplify. 18·2·518·2·5
365365
Step-by-step guide to multiplying and simplifying radical expressions, from (6√2)(3√10) to 36√5.

ⓑ

(−543)(−463)(−543)(−463)
Multiply using the Product Property. 2024320243
Simplify the radical. 2083·332083·33
Simplify. 20·2·3320·2·33
40334033
Step-by-step multiplication and simplification of cube root expressions.

Try It 8.79

Simplify: ⓐ (32)(230)(32)(230) ⓑ (2183)(−363).(2183)(−363).

Try It 8.80

Simplify: ⓐ (33)(36)(33)(36) ⓑ (−493)(363).(−493)(363).

We follow the same procedures when there are variables in the radicands.

Example 8.41

Simplify: ⓐ (106p3)(43p)(106p3)(43p) ⓑ (220y24)(328y34).(220y24)(328y34).

Solution

ⓐ

(106p3)(43p)(106p3)(43p)
Multiply. 4018p44018p4
Simplify the radical. 409p4·2409p4·2
Simplify. 40·3p2·240·3p2·2
120p22120p22
This table illustrates the step-by-step process of multiplying and simplifying a radical expression, showing the intermediate results.

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

(220y24)(328y34)(220y24)(328y34)
Multiply. 64·5·4·7y5464·5·4·7y54
Simplify the radical. 616y44·35y4616y44·35y4
Simplify. 6·2y35y46·2y35y4
Multiply. 12y35y412y35y4
Step-by-step solution for multiplying and simplifying radical expressions involving fourth roots.

Try It 8.81

Simplify: ⓐ (66x2)(830x4)(66x2)(830x4) ⓑ (−412y34)(−8y34).(−412y34)(−8y34).

Try It 8.82

Simplify: ⓐ (26y4)(1230y)(26y4)(1230y) ⓑ (−49a34)(327a24).(−49a34)(327a24).

Use Polynomial Multiplication to Multiply Radical Expressions

In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals when possible.

Example 8.42

Simplify: ⓐ 6(2+18)6(2+18) ⓑ 93(5−183).93(5−183).

Solution

ⓐ

6(2+18)6(2+18)
Multiply. 12+10812+108
Simplify. 4·3+36·34·3+36·3
Simplify. 23+6323+63
Combine like radicals. 8383
This table outlines the sequential steps involved in simplifying a radical expression, from initial multiplication to combining radicals.

ⓑ

93(5−183)93(5−183)
Distribute. 593−1623593−1623
Simplify. 593−273·63593−273·63
Simplify. 593−363593−363
This table illustrates the step-by-step simplification of an algebraic expression involving cube roots through distribution and simplification.

Try It 8.83

Simplify: ⓐ 6(1+36)6(1+36) ⓑ 43(−2−63).43(−2−63).

Try It 8.84

Simplify: ⓐ 8(2−58)8(2−58) ⓑ 33(−93−63).33(−93−63).

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Example 8.43

Simplify: ⓐ (3−27)(4−27)(3−27)(4−27) ⓑ (x3−2)(x3+4).(x3−2)(x3+4).

Solution

ⓐ

(3−27)(4−27)(3−27)(4−27)
Multiply. 12−67−87+4·712−67−87+4·7
Simplify. 12−67−87+2812−67−87+28
Combine like terms. 40−14740−147
This table illustrates the step-by-step multiplication and simplification of two binomial expressions containing square roots.

ⓑ

(x3−2)(x3+4)(x3−2)(x3+4)
Multiply. x23+4x3−2x3−8x23+4x3−2x3−8
Combine like terms. x23+2x3−8x23+2x3−8
This table demonstrates the step-by-step multiplication and simplification of an algebraic expression involving cube roots.

Try It 8.85

Simplify: ⓐ (6−37)(3+47)(6−37)(3+47) ⓑ (x3−2)(x3−3).(x3−2)(x3−3).

Try It 8.86

Simplify: ⓐ (2−311)(4−11)(2−311)(4−11) ⓑ (x3+1)(x3+3).(x3+1)(x3+3).

Example 8.44

Simplify: (32−5)(2+45).(32−5)(2+45).

Solution

(32−5)(2+45)(32−5)(2+45)
Multiply. 3·2+1210−10−4·53·2+1210−10−4·5
Simplify. 6+1210−10−206+1210−10−20
Combine like terms. −14+1110−14+1110
Step-by-step multiplication and simplification of two radical expressions, showing the progression from initial problem to final solution.

Try It 8.87

Simplify: (53−7)(3+27)(53−7)(3+27)

Try It 8.88

Simplify: (6−38)(26+8)(6−38)(26+8)

Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.

Special Products

Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2Binomial SquaresProduct of Conjugates(a+b)2=a2+2ab+b2(a+b)(a−b)=a2−b2(a−b)2=a2−2ab+b2

We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.

Example 8.45

Simplify: ⓐ (2+3)2(2+3)2 ⓑ (4−25)2.(4−25)2.

Solution

Be sure to include the 2ab2ab term when squaring a binomial.

ⓐ

Two mathematical expressions are shown: (a + b)' in red text, representing the square of a sum, and directly below it, (2 + sqrt(3))', a specific example of the same algebraic form.
Multiply, using the Product of Binomial Squares Pattern. Algebraic identity for a perfect square: a^2 + 2ab + b^2, with an example where a=2 and b=sqrt(3).
Simplify. A mathematical expression reads '4 + 4 square root of 3 + 3' in black text against a white background, representing an algebraic sum involving an integer and a surd.
Combine like terms. A mathematical expression '7 + 4√3' is displayed on a white background.



ⓑ

The image displays two algebraic expressions related to squaring a difference. The first expression, in red, is (a - b)^2. Below it, in black, is the specific numerical instance (4 - 2√5)^2.
Multiply, using the Product of Binomial Squares Pattern. Illustration of the perfect square trinomial identity (a-b)^2, showing a^2 - 2ab + b^2 and its numerical application with a=4 and b=2sqrt(5).
Simplify. A mathematical expression reads '16 - 16√5 + 4 × 5' against a white background.
The image shows the mathematical expression 16 - 16√5 + 20.
Combine like terms. The image shows the mathematical expression 36 - 16√5 in a bold, black font against a white background.

Try It 8.89

Simplify: ⓐ (10+2)2(10+2)2 ⓑ (1+36)2.(1+36)2.

Try It 8.90

Simplify: ⓐ (6−5)2(6−5)2 ⓑ (9−210)2.(9−210)2.

In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.

Example 8.46

Simplify: (5−23)(5+23).(5−23)(5+23).

Solution

This image displays the difference of squares formula, (a-b)(a+b), and an application of it with radical numbers: (5 - 2√3)(5 + 2√3).
Multiply, using the Product of Conjugates Pattern. The image displays a mathematical problem with two lines of expressions. The first line, in red text, shows the difference of squares formula, a^2 - b^2. Below it, in black text, is the specific calculation 5^2 - (2√3)^2.
Simplify. A mathematical expression '25 - 4 ⋅ 3' is shown in black text on a white background.
The number '13' is visible on the left side of the image.

Try It 8.91

Simplify: (3−25)(3+25)(3−25)(3+25)

Try It 8.92

Simplify: (4+57)(4−57).(4+57)(4−57).

Media

Access these online resources for additional instruction and practice with adding, subtracting, and multiplying radical expressions.

Section 8.4 Exercises

Practice Makes Perfect

Add and Subtract Radical Expressions

In the following exercises, simplify.

165.

ⓐ 82−5282−52 ⓑ 5m3+2m35m3+2m3 ⓒ 8m4−2m48m4−2m4

166.

ⓐ 72−3272−32 ⓑ 7p3+2p37p3+2p3 ⓒ 5x3−3x35x3−3x3

167.

ⓐ 35+6535+65 ⓑ 9a3+3a39a3+3a3 ⓒ 52z4+2z452z4+2z4

168.

ⓐ 45+8545+85 ⓑ m3−4m3m3−4m3 ⓒ n+3nn+3n

169.

ⓐ 32a−42a+52a32a−42a+52a ⓑ 53ab4−33ab4−23ab453ab4−33ab4−23ab4

170.

ⓐ 11b−511b+311b11b−511b+311b ⓑ 811cd4+511cd4−911cd4811cd4+511cd4−911cd4

171.

ⓐ 83c+23c−93c83c+23c−93c ⓑ 24pq3−54pq3+44pq324pq3−54pq3+44pq3

172.

ⓐ 35d+85d−115d35d+85d−115d ⓑ 112rs3−92rs3+32rs3112rs3−92rs3+32rs3

173.

ⓐ 27−7527−75 ⓑ 403−3203403−3203 ⓒ 12324+23162412324+231624

174.

ⓐ 72−9872−98 ⓑ 243+813243+813 ⓒ 12804−23405412804−234054

175.

ⓐ 48+2748+27 ⓑ 543+1283543+1283 ⓒ 654−32804654−32804

176.

ⓐ 45+8045+80 ⓑ 813−1923813−1923 ⓒ 52804+73405452804+734054

177.

ⓐ 72a5−50a572a5−50a5 ⓑ 980p44−6405p44980p44−6405p44

178.

ⓐ 48b5−75b548b5−75b5 ⓑ 864q63−3125q63864q63−3125q63

179.

ⓐ 80c7−20c780c7−20c7 ⓑ 2162r104+432r1042162r104+432r104

180.

ⓐ 96d9−24d996d9−24d9 ⓑ 5243s64+23s645243s64+23s64

181.

3 128 y 2 + 4 y 162 − 8 98 y 2 3 128 y 2 + 4 y 162 − 8 98 y 2

182.

3 75 y 2 + 8 y 48 − 300 y 2 3 75 y 2 + 8 y 48 − 300 y 2

Multiply Radical Expressions

In the following exercises, simplify.

183.

ⓐ (−23)(318)(−23)(318) ⓑ (843)(−4183)(843)(−4183)

184.

ⓐ (−45)(510)(−45)(510) ⓑ (−293)(793)(−293)(793)

185.

ⓐ (56)(−12)(56)(−12) ⓑ (−2184)(−94)(−2184)(−94)

186.

ⓐ (−27)(−214)(−27)(−214) ⓑ (−384)(−564)(−384)(−564)

187.

ⓐ (412z3)(39z)(412z3)(39z) ⓑ (53x33)(318x33)(53x33)(318x33)

188.

ⓐ (32x3)(718x2)(32x3)(718x2) ⓑ (−620a23)(−216a33)(−620a23)(−216a33)

189.

ⓐ (−27z3)(314z8)(−27z3)(314z8) ⓑ (28y24)(−212y34)(28y24)(−212y34)

190.

ⓐ (42k5)(−332k6)(42k5)(−332k6) ⓑ (−6b34)(38b34)(−6b34)(38b34)

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

191.

ⓐ 7(5+27)7(5+27) ⓑ 63(4+183)63(4+183)

192.

ⓐ 11(8+411)11(8+411) ⓑ 33(93+183)33(93+183)

193.

ⓐ 11(−3+411)11(−3+411) ⓑ 34(544+184)34(544+184)

194.

ⓐ 2(−5+92)2(−5+92) ⓑ 24(124+244)24(124+244)

195.

( 7 + 3 ) ( 9 − 3 ) ( 7 + 3 ) ( 9 − 3 )

196.

( 8 − 2 ) ( 3 + 2 ) ( 8 − 2 ) ( 3 + 2 )

197.

ⓐ (9−32)(6+42)(9−32)(6+42) ⓑ (x3−3)(x3+1)(x3−3)(x3+1)

198.

ⓐ (3−27)(5−47)(3−27)(5−47) ⓑ (x3−5)(x3−3)(x3−5)(x3−3)

199.

ⓐ (1+310)(5−210)(1+310)(5−210) ⓑ (2x3+6)(x3+1)(2x3+6)(x3+1)

200.

ⓐ (7−25)(4+95)(7−25)(4+95) ⓑ (3x3+2)(x3−2)(3x3+2)(x3−2)

201.

( 3 + 10 ) ( 3 + 2 10 ) ( 3 + 10 ) ( 3 + 2 10 )

202.

( 11 + 5 ) ( 11 + 6 5 ) ( 11 + 5 ) ( 11 + 6 5 )

203.

( 2 7 − 5 11 ) ( 4 7 + 9 11 ) ( 2 7 − 5 11 ) ( 4 7 + 9 11 )

204.

( 4 6 + 7 13 ) ( 8 6 − 3 13 ) ( 4 6 + 7 13 ) ( 8 6 − 3 13 )

205.

ⓐ (3+5)2(3+5)2 ⓑ (2−53)2(2−53)2

206.

ⓐ (4+11)2(4+11)2 ⓑ (3−25)2(3−25)2

207.

ⓐ (9−6)2(9−6)2 ⓑ (10+37)2(10+37)2

208.

ⓐ (5−10)2(5−10)2 ⓑ (8+32)2(8+32)2

209.

( 4 + 2 ) ( 4 − 2 ) ( 4 + 2 ) ( 4 − 2 )

210.

( 7 + 10 ) ( 7 − 10 ) ( 7 + 10 ) ( 7 − 10 )

211.

( 4 + 9 3 ) ( 4 − 9 3 ) ( 4 + 9 3 ) ( 4 − 9 3 )

212.

( 1 + 8 2 ) ( 1 − 8 2 ) ( 1 + 8 2 ) ( 1 − 8 2 )

213.

( 12 − 5 5 ) ( 12 + 5 5 ) ( 12 − 5 5 ) ( 12 + 5 5 )

214.

( 9 − 4 3 ) ( 9 + 4 3 ) ( 9 − 4 3 ) ( 9 + 4 3 )

215.

( 3 x 3 + 2 ) ( 3 x 3 − 2 ) ( 3 x 3 + 2 ) ( 3 x 3 − 2 )

216.

( 4 x 3 + 3 ) ( 4 x 3 − 3 ) ( 4 x 3 + 3 ) ( 4 x 3 − 3 )

Mixed Practice

217.

2 3 27 + 3 4 48 2 3 27 + 3 4 48

218.

175 k 4 − 63 k 4 175 k 4 − 63 k 4

219.

5 6 162 + 3 16 128 5 6 162 + 3 16 128

220.

24 3 + / 81 3 24 3 + / 81 3

221.

1 2 80 4 − 2 3 405 4 1 2 80 4 − 2 3 405 4

222.

8 13 4 − 4 13 4 − 3 13 4 8 13 4 − 4 13 4 − 3 13 4

223.

5 12 c 4 − 3 27 c 6 5 12 c 4 − 3 27 c 6

224.

80 a 5 − 45 a 5 80 a 5 − 45 a 5

225.

3 5 75 − 1 4 48 3 5 75 − 1 4 48

226.

21 9 3 − 2 9 3 21 9 3 − 2 9 3

227.

8 64 q 6 3 − 3 125 q 6 3 8 64 q 6 3 − 3 125 q 6 3

228.

11 11 − 10 11 11 11 − 10 11

229.

3 · 21 3 · 21

230.

( 4 6 ) ( − 18 ) ( 4 6 ) ( − 18 )

231.

( 7 4 3 ) ( −3 18 3 ) ( 7 4 3 ) ( −3 18 3 )

232.

( 4 12 x 5 ) ( 2 6 x 3 ) ( 4 12 x 5 ) ( 2 6 x 3 )

233.

( 29 ) 2 ( 29 ) 2

234.

( −4 17 ) ( −3 17 ) ( −4 17 ) ( −3 17 )

235.

( −4 + 17 ) ( −3 + 17 ) ( −4 + 17 ) ( −3 + 17 )

236.

( 3 8 a 2 4 ) ( 12 a 3 4 ) ( 3 8 a 2 4 ) ( 12 a 3 4 )

237.

( 6 − 3 2 ) 2 ( 6 − 3 2 ) 2

238.

3 ( 4 − 3 3 ) 3 ( 4 − 3 3 )

239.

3 3 ( 2 9 3 + 18 3 ) 3 3 ( 2 9 3 + 18 3 )

240.

( 6 + 3 ) ( 6 + 6 3 ) ( 6 + 3 ) ( 6 + 6 3 )

Writing Exercises

241.

Explain when a radical expression is in simplest form.

242.

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.

243.


ⓐ Explain why (−n)2(−n)2 is always non-negative, for n≥0.n≥0.
ⓑ Explain why −(n)2−(n)2 is always non-positive, for n≥0.n≥0.

244.

Use the binomial square pattern to simplify (3+2)2.(3+2)2. Explain all your steps.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “add and subtract radical expressions.”, “ multiply radical expressions”, and “use polynomial multiplication to multiply radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?