Document created by South Campus Library Learning Commons 11/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed.
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MAT1033-3
Simplifying Radicals
BACKGROUND When used in reference to radicals, to “simplify” will mean to remove from the radicals any perfect roots that occur as factors. The following are rules governing this simplification process.
RULES 1. One factor outside a radical equals !" !"#$ !" !ℎ! !"#$% !"#!$%
Examples:
a. 2 = 2 ∗ 2 b. 2 = 2 ∗ 2 ∗ 2!
c. 2 = 2 ∗ 2 ∗ 2 ∗ 2!
d. 2 5! = 2 ∗ 2 ∗ 2 ∗ 5!
e. 2!!! = 2 ∗ ! ∗ ! ∗ ! ∗ ! ∗ !! = ! 2!!!
2. Only factors may be removed!
Examples:
!. !! = !!! = ! !
!. !! + !! = no reduction because not factorable
!. !! + 2! + 1 = ! + 1 ! = ! + 1 3. Multiplying radicals: Possible only if same index
Examples:
a. 2!! * 3!! = 2! ∗ 3!! = 6!!!
!"#$%$&!!"#"$ !"#$%&#' ! !"#$%&'" !"#$%&#%.
b. 2!! * −3!! !"##$% !" !"#$%&#%'( (!"# !"#$%&"' !"" !"#$!%#&' !"#$%&')
Document created by South Campus Library Learning Commons 11/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed.
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4. Dividing radicals: Possible only if same index
Examples:
!. !!! !
!! ! = !!!
!!
!= 3!!! = ! 3!
Also note the reverse:
!. 27!33!
3= 27 !33
3!3 = 3!3!3
5. Multiply any radical that needs removing by another (same index) that will
complete the set the index requires.
Examples:
a. Remove the radical from the denominator in the following
expression: !!
1!∗
!!=
!!
Note: We just multiplied the expression by 1 (!!), yet were able to put
the radical in the numerator, where it can be more easily handled.
b. Remove the radical from the denominator in the following expression:
3!2 !!
3!2 !! ∗
!!!
!!! =3! !!!
2! =3 !!!
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Note: This rule is also known as “Rationalization”. Usually performed on a radical in the denominator, it is the preferred choice for removing unwanted radicals.
Document created by South Campus Library Learning Commons 11/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed.
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EXERCISES Simplify
1. 16
2. 64
3. 49
4. 144
5. 32
6. 50
7. 8
8. 12
9. 6 18
10. −3 48
11. 5 40
12. 2 28
13. 15
14. 21
15. 29
16. 13
17. −9 72
18. 11 80
19. 45
20. 225
21. 0
22. 210
23. 6 128
24. 9 288
25. 105
26. 55
27. 900
28. 300
29. 5 180
30. 7 98
31. 250
32. 120
33. 96
34. 160
35. 324
36. 444
37. !!
38. !!"
39. !!"
40. !!!
41. !!"
42. !!!!
43. !!"!!
44. (81(! − !)!
45. 4 20!!!!
46. 9(! − 2)!
47. !! + 4! + 4
48. !! − 8! + 16
49. !! + 6! + 9
50. ! !!!!
Document created by South Campus Library Learning Commons 11/01/09. Permission to copy and use is granted for educational use provided this copyright label is displayed.
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SOLUTIONS 1. 16 = ±4
2. 64 = ±8
3. 49 = ±7
4. 144 = ±12
5. 32 = 4√2
6. 50 = 5√2
7. 8 = 2√2
8. 12 = 2√3
9. 6 18 = 18√2
10. −3 48 = -12√3
11. 5 40 = 10√10
12. 2 28 = 4√7
13. 15 = √15
14. 21 = √21
15. 29 = √29
16. 13 = √13
17. −9 72 = -54√2
18. 11 80 = 44√5
19. 45 = 3√5
20. 225 = ±15
21. 0 = 0
22. 210 = √210
23. 6 128 = 48√2
24. 9 288 = 108√2
25. 105 = √105
26. 55= √55
27. 900= ±30
28. 300 = 10√3
29. 5 180= 30√5
30. 7 98= 49√2
31. 250= 5√10
32. 120= 2√30
33. 96 = 4√6
34. 160 = 4√10
35. 324 = ±18
36. 444= 2√111
37. !! = x3
38. !!" = x6
39. !!" = y7√y
40. !!! = y5√y
41. !!" = a10
42. !!!! = x2y2
43. !!"!! = x6y4
44. (81(! − !)! = 9x2-
18xy+9y2
45. 4 20!!!! = 8a2b3√5b
46. 9(! − 2)! = 3(x-2)4
47. !! + 4! + 4 = x + 2
48. !! − 8! + 16 = b - 4
49. !! + 6! + 9 = a + 3
50. ! !!!! = x3y
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