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.  

1  

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.  

2  

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 !!!

2

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.  

3  

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.  

4  

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