9.3 Add And Subtract Square Roots - Elementary Algebra 2e

9.3 Add and Subtract Square Roots

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

• Add and subtract like square roots
• Add and subtract square roots that need simplification

Before you get started, take this readiness quiz.

Add: ⓐ 3x+9x3x+9x ⓑ 5m+5n5m+5n. If you missed this problem, review Example 1.24.

Simplify: 50x350x3. If you missed this problem, review Example 9.16.

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify 2+72+7 in this way:

2+7Add inside the radical.9Simplify.32+7Add inside the radical.9Simplify.3

So if we have to add 2+72+7, we must not combine them into one radical.

2+7≠2+72+7≠2+7

Trying to add square roots with different radicands is like trying to add unlike terms.

But, just like we can addx+x,we can add3+3.x+x=2x3+3=23But, just like we can addx+x,we can add3+3.x+x=2x3+3=23

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

Square roots with the same radicand are called like square roots.

We add and subtract like square roots in the same way we add and subtract like terms. We know that 3x+8x3x+8x is 11x11x. Similarly we add 3x+8x3x+8x and the result is 11x.11x.

Add and Subtract Like Square Roots

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

Simplify: 22−7222−72.

Solution

22−7222−72
Since the radicals are like, we subtract the coefficients.	−52−52

Simplify: 82−9282−92.

Simplify: 53−9353−93.

Simplify: 3y+4y3y+4y.

Solution

3y+4y3y+4y
Since the radicals are like, we add the coefficients.	7y7y

Simplify: 2x+7x2x+7x.

Simplify: 5u+3u5u+3u.

Simplify: 4x−2y4x−2y.

Solution

4x−2y4x−2y
Since the radicals are not like, we cannotsubtract them. We leave the expression as is.	4x−2y4x−2y

Simplify: 7p−6q7p−6q.

Simplify: 6a−3b6a−3b.

Simplify: 513+413+213513+413+213.

Solution

513+413+213513+413+213
Since the radicals are like, we add the coefficients.	11131113

Simplify: 411+211+311411+211+311.

Simplify: 610+210+310610+210+310.

Simplify: 26−66+3326−66+33.

Solution

26−66+3326−66+33
Since the first two radicals are like, wesubtract their coefficients.	−46+33−46+33

Simplify: 55−45+2655−45+26.

Simplify: 37−87+2537−87+25.

Simplify: 25n−65n+45n25n−65n+45n.

Solution

25n−65n+45n25n−65n+45n
Since the radicals are like, we combine them.	05n05n
Simplify.	0

Simplify: 7x−77x+47x7x−77x+47x.

Simplify: 43y−73y+23y43y−73y+23y.

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

Simplify: 3xy+53xy−43xy3xy+53xy−43xy.

Solution

3xy+53xy−43xy3xy+53xy−43xy
Since the radicals are like, we combine them.	23xy23xy

Simplify: 5xy+45xy−75xy5xy+45xy−75xy.

Simplify: 37mn+7mn−47mn37mn+7mn−47mn.

Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Simplify: 20+3520+35.

Solution

20+3520+35
Simplify the radicals, when possible.	4·5+354·5+35
25+3525+35
Combine the like radicals.	5555

Simplify: 18+6218+62.

Simplify: 27+4327+43.

Simplify: 48−7548−75.

Solution

48−7548−75
Simplify the radicals.	16·3−25·316·3−25·3
43−5343−53
Combine the like radicals.	−3−3

Simplify: 32−1832−18.

Simplify: 20−4520−45.

Just like we use the Associative Property of Multiplication to simplify 5(3x)5(3x) and get 15x15x, we can simplify 5(3x)5(3x) and get 15x15x. We will use the Associative Property to do this in the next example.

Simplify: 518−28518−28.

Solution

518−28518−28
Simplify the radicals.	5·9·2−2·4·25·9·2−2·4·2
5·3·2−2·2·25·3·2−2·2·2
152−42152−42
Combine the like radicals.	112112

Simplify: 427−312427−312.

Simplify: 320−745320−745.

Simplify: 34192−5610834192−56108.

Solution

34192−5610834192−56108
Simplify the radicals.	3464·3−5636·33464·3−5636·3
34·8·3−56·6·334·8·3−56·6·3
63−5363−53
Combine the like radicals.	33

Simplify: 23108−5714723108−57147.

Simplify: 35200−3412835200−34128.

Simplify: 2348−34122348−3412.

Solution

2348−34122348−3412
Simplify the radicals.	2316·3−344·32316·3−344·3
23·4·3−34·2·323·4·3−34·2·3
833−323833−323
Find a common denominator to subtract thecoefficients of the like radicals.	1663−9631663−963
Simplify.	763763

Simplify: 2532−1382532−138.

Simplify: 1380−141251380−14125.

In the next example, we will remove constant and variable factors from the square roots.

Simplify: 18n5−32n518n5−32n5.

Solution

18n5−32n518n5−32n5
Simplify the radicals.	9n4·2n−16n4·2n9n4·2n−16n4·2n
3n22n−4n22n3n22n−4n22n
Combine the like radicals.	−n22n−n22n

Simplify: 32m7−50m732m7−50m7.

Simplify: 27p3−48p327p3−48p3.

Simplify: 950m2−648m2950m2−648m2.

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.

Simplify: 532x2−348x2532x2−348x2.

Simplify: 748y2−472y2748y2−472y2.

Simplify: 28x2−5x32+518x228x2−5x32+518x2.

Solution

28x2−5x32+518x228x2−5x32+518x2
Simplify the radicals.	24x2·2−5x16·2+59x2·224x2·2−5x16·2+59x2·2
2·2x·2−5x·4·2+5·3x·22·2x·2−5x·4·2+5·3x·2
4x2−20x2+15x24x2−20x2+15x2
Combine the like radicals.	−x2−x2

Simplify: 312x2−2x48+427x2312x2−2x48+427x2.

Simplify: 318x2−6x32+250x2318x2−6x32+250x2.

Access this online resource for additional instruction and practice with the adding and subtracting square roots.

• Adding/Subtracting Square Roots

Section 9.3 Exercises

Practice Makes Perfect

Add and Subtract Like Square Roots

In the following exercises, simplify.

145.

8 2 − 5 2 8 2 − 5 2

146.

7 2 − 3 2 7 2 − 3 2

147.

3 5 + 6 5 3 5 + 6 5

148.

4 5 + 8 5 4 5 + 8 5

149.

9 7 − 10 7 9 7 − 10 7

150.

11 7 − 12 7 11 7 − 12 7

151.

7 y + 2 y 7 y + 2 y

152.

9 n + 3 n 9 n + 3 n

153.

a − 4 a a − 4 a

154.

b − 6 b b − 6 b

155.

5 c + 2 c 5 c + 2 c

156.

7 d + 2 d 7 d + 2 d

157.

8 a − 2 b 8 a − 2 b

158.

5 c − 3 d 5 c − 3 d

159.

5 m + n 5 m + n

160.

n + 3 p n + 3 p

161.

8 7 + 2 7 + 3 7 8 7 + 2 7 + 3 7

162.

6 5 + 3 5 + 5 6 5 + 3 5 + 5

163.

3 11 + 2 11 − 8 11 3 11 + 2 11 − 8 11

164.

2 15 + 5 15 − 9 15 2 15 + 5 15 − 9 15

165.

3 3 − 8 3 + 7 5 3 3 − 8 3 + 7 5

166.

5 7 − 8 7 + 6 3 5 7 − 8 7 + 6 3

167.

6 2 + 2 2 − 3 5 6 2 + 2 2 − 3 5

168.

7 5 + 5 − 8 10 7 5 + 5 − 8 10

169.

3 2 a − 4 2 a + 5 2 a 3 2 a − 4 2 a + 5 2 a

170.

11 b − 5 11 b + 3 11 b 11 b − 5 11 b + 3 11 b

171.

8 3 c + 2 3 c − 9 3 c 8 3 c + 2 3 c − 9 3 c

172.

3 5 d + 8 5 d − 11 5 d 3 5 d + 8 5 d − 11 5 d

173.

5 3 a b + 3 a b − 2 3 a b 5 3 a b + 3 a b − 2 3 a b

174.

8 11 c d + 5 11 c d − 9 11 c d 8 11 c d + 5 11 c d − 9 11 c d

175.

2 p q − 5 p q + 4 p q 2 p q − 5 p q + 4 p q

176.

11 2 r s − 9 2 r s + 3 2 r s 11 2 r s − 9 2 r s + 3 2 r s

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

177.

50 + 4 2 50 + 4 2

178.

48 + 2 3 48 + 2 3

179.

80 − 3 5 80 − 3 5

180.

28 − 4 7 28 − 4 7

181.

27 − 75 27 − 75

182.

72 − 98 72 − 98

183.

48 + 27 48 + 27

184.

45 + 80 45 + 80

185.

2 50 − 3 72 2 50 − 3 72

186.

3 98 − 128 3 98 − 128

187.

2 12 + 3 48 2 12 + 3 48

188.

4 75 + 2 108 4 75 + 2 108

189.

2 3 72 + 1 5 50 2 3 72 + 1 5 50

190.

2 5 75 + 3 4 48 2 5 75 + 3 4 48

191.

1 2 20 − 2 3 45 1 2 20 − 2 3 45

192.

2 3 54 − 3 4 96 2 3 54 − 3 4 96

193.

1 6 27 − 3 8 48 1 6 27 − 3 8 48

194.

1 8 32 − 1 10 50 1 8 32 − 1 10 50

195.

1 4 98 − 1 3 128 1 4 98 − 1 3 128

196.

1 3 24 + 1 4 54 1 3 24 + 1 4 54

197.

72 a 5 − 50 a 5 72 a 5 − 50 a 5

198.

48 b 5 − 75 b 5 48 b 5 − 75 b 5

199.

80 c 7 − 20 c 7 80 c 7 − 20 c 7

200.

96 d 9 − 24 d 9 96 d 9 − 24 d 9

201.

9 80 p 4 − 6 98 p 4 9 80 p 4 − 6 98 p 4

202.

8 72 q 6 − 3 75 q 6 8 72 q 6 − 3 75 q 6

203.

2 50 r 8 + 4 54 r 8 2 50 r 8 + 4 54 r 8

204.

5 27 s 6 + 2 20 s 6 5 27 s 6 + 2 20 s 6

205.

3 20 x 2 − 4 45 x 2 + 5 x 80 3 20 x 2 − 4 45 x 2 + 5 x 80

206.

2 28 x 2 − 63 x 2 + 6 x 7 2 28 x 2 − 63 x 2 + 6 x 7

207.

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

208.

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

Mixed Practice

209.

2 8 + 6 8 − 5 8 2 8 + 6 8 − 5 8

210.

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

211.

175 k 4 − 63 k 4 175 k 4 − 63 k 4

212.

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

213.

2 363 − 2 300 2 363 − 2 300

214.

150 + 4 6 150 + 4 6

215.

9 2 − 8 2 9 2 − 8 2

216.

5 x − 8 y 5 x − 8 y

217.

8 13 − 4 13 − 3 13 8 13 − 4 13 − 3 13

218.

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

219.

80 a 5 − 45 a 5 80 a 5 − 45 a 5

220.

3 5 75 − 1 4 48 3 5 75 − 1 4 48

221.

21 19 − 2 19 21 19 − 2 19

222.

500 + 405 500 + 405

223.

5 6 27 + 5 8 48 5 6 27 + 5 8 48

224.

11 11 − 10 11 11 11 − 10 11

225.

75 − 108 75 − 108

226.

2 98 − 4 72 2 98 − 4 72

227.

4 24 x 2 − 54 x 2 + 3 x 6 4 24 x 2 − 54 x 2 + 3 x 6

228.

8 80 y 6 − 6 48 y 6 8 80 y 6 − 6 48 y 6

Everyday Math

229.

A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression 9(82)+8(22)9(82)+8(22). (Round to the nearest tenth of an inch.)

230.

Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression 412+88+104412+88+104 to determine the width of the wall. (Round to the nearest tenth of an inch.)

Writing Exercises

231.

Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

232.

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.

Self Check

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?