4.5E: Exercises
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- Feb 26, 2021
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Practice Makes Perfect
Exercise A: add and subtract radical expressions
In the following exercises, simplify. Assume all variables are greater than or equal to zero so that absolute values are not needed.
- a. 8√2−5√2 b. 53√m+23√m c. 84√m−24√n
- a. 7√2−3√2 b. 73√p+23√p c. 53√x−33√x
- a. 3√5+6√5 b. 93√a+33√a c. 54√2z+4√2z
- a. 4√5+8√5 b. 3√m−43√m c. √n+3√n
- a. 3√2a−4√2a+5√2a b. 54√3ab−34√3ab−24√3ab
- a. √11b−5√11b+3√11b b. 84√11cd+54√11cd−94√11cd
- a. 8√3c+2√3c−9√3c b. 23√4pq−53√4pq+43√4pq
- a. 3√5d+8√5d−11√5d b. 113√2rs−93√2rs+33√2rs
- a. √27−√75 b. 3√40−3√320 c. 124√32+234√162
- a. √72−√98 b. 3√24+3√81 c. 124√80−234√405
- a. √48+√27 b. 3√54+3√128 c. 64√5−324√320
- a. √45+√80 b. 3√81−3√192 c. 524√80+734√405
- a. √72a5−√50a5 b. 94√80p4−64√405p4
- a. √48b5−√75b5 b. 83√64q6−33√125q6
- a. √80c7−√20c7 b. 24√162r10+44√32r10
- a. √96d9−√24d9 b. 54√243s6+24√3s6
- 3√128y2+4y√162−8√98y2
- 3√75y2+8y√48−√300y2
- Answer
-
1. a. 3√2 b. 73√m c. 64√m
3. a. 9√5 b. 123√a c. 64√2z
5. a. 4√2a b. 0
7. a. √3c b. 3√4pq
9. a. −2√3 b. −23√5 c. 34√2
11. a. 7√3 b. 73√2 c. 34√5
13. a. a2√2a b. 0
15. a. 2c3√5c b. 14r24√2r2
17. 4y√2
Exercise B: multiply radical expressions
In the following exercises, simplify.
-
- (−2√3)(3√18)
- (83√4)(−43√18)
-
- (−4√5)(5√10)
- (−23√9)(73√9)
-
- (5√6)(−√12)
- (−24√18)(−4√9)
-
- (−2√7)(−2√14)
- (−34√8)(−54√6)
-
- (4√12z3)(3√9z)
- (53√3x3)(33√18x3)
-
- (3√2x3)(7√18x2)
- (−63√20a2)(−23√16a3)
-
- (−2√7z3)(3√14z8)
- (24√8y2)(−24√12y3)
-
- (4√2k5)(−3√32k6)
- (−4√6b3)(34√8b3)
- Answer
-
1.
- −18√6
- −643√9
3.
- −30√2
- 64√2
5.
- 72z2√3
- 45x23√2
7.
- −42z5√2z
- −8y4√6y
Exercise C: use polynomial multiplication to multiply radical expressions
In the following exercises, multiply.
-
- √7(5+2√7)
- 3√6(4+3√18)
-
- √11(8+4√11)
- 3√3(3√9+3√18)
-
- √11(−3+4√11)
- 4√3(4√54+4√18)
-
- √2(−5+9√2)
- 4√2(4√12+4√24)
- (7+√3)(9−√3)
- (8−√2)(3+√2)
-
- (9−3√2)(6+4√2)
- (3√x−3)(3√x+1)
-
- (3−2√7)(5−4√7)
- (3√x−5)(3√x−3)
-
- (1+3√10)(5−2√10)
- (23√x+6)(3√x+1)
-
- (7−2√5)(4+9√5)
- (33√x+2)(3√x−2)
- (√3+√10)(√3+2√10)
- (√11+√5)(√11+6√5)
- (2√7−5√11)(4√7+9√11)
- (4√6+7√13)(8√6−3√13)
-
- (3+√5)2
- (2−5√3)2
-
- (4+√11)2
- (3−2√5)2
-
- (9−√6)2
- (10+3√7)2
-
- (5−√10)2
- (8+3√2)2
- (4+√2)(4−√2)
- (7+√10)(7−√10)
- (4+9√3)(4−9√3)
- (1+8√2)(1−8√2)
- (12−5√5)(12+5√5)
- (9−4√3)(9+4√3)
- (3√3x+2)(3√3x−2)
- (3√4x+3)(3√4x−3)
- Answer
-
1.
- 14+5√7
- 43√6+33√4
3.
- 44−3√11
- 34√2+4√54
5. 60+2√3
7.
- 30+18√2
- 3√x2−23√x−3
9.
- −54+13√10
- 23√x2+83√x+6
11. 23+3√30
13. −439−2√77
15.
- 14+6√5
- 79−20√3
17.
- 87−18√6
- 163+60√7
19. 14
21. −227
23. 19
25. 3√9x2−4
Exercise D: mixed practice
- 23√27+34√48
- √175k4−√63k4
- 56√162+316√128
- 3√24+3√81
- 124√80−234√405
- 84√13−44√13−34√13
- 5√12c4−3√27c6
- √80a5−√45a5
- 35√75−14√48
- 213√9−23√9
- 83√64q6−33√125q6
- 11√11−10√11
- √3⋅√21
- (4√6)(−√18)
- (73√4)(−33√18)
- (4√12x5)(2√6x3)
- (√29)2
- (−4√17)(−3√17)
- (−4+√17)(−3+√17)
- (34√8a2)(4√12a3)
- (6−3√2)2
- √3(4−3√3)
- 3√3(23√9+3√18)
- (√6+√3)(√6+6√3)
- Answer
-
1. 5√3
3. 9√2
5. −4√5
7. 10c2√3−9c3√3
9. 2√3
11. 17q2
13. 3√7
15. −423√9
17. 29
19. 29−7√17
21. 72−36√2
23. 6+33√2
Exercise E: writing exercises
- Explain when a radical expression is in simplest form.
- 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.
-
- Explain why (−√n)2 is always non-negative, for n≥0.
- Explain why −(√n)2 is always non-positive, for n≥0.
- Use the binomial square pattern to simplify (3+√2)2. Explain all your steps.
- Answer
-
1. Answers will vary
3. Answers will vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. 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?