1. 5-1: Radicals & Rational
Exponents
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define and apply rational
exponents
•Perform arithmetic
operations with radicals
• Simplify expressions with
radicals or rational exponents

2. Definition
A rational number is any
number that can be written
in the form:
, 0
m
n
n
≠

3. Definition
A rational exponent is any
number that can be written
in the form:
, 0
m
n
b n ≠

4. Definition
m
n mn
b b=
Rational exponent form
Radical form

5. Definition
m
n
b b

6. Definition
n m
b
n is the index of the radical.
If the index is left blank, it is
understood to be 2.
m is the exponent of the
radical.

7. Example
Evaluate without using a
calculator:
1
2
81
2
3
27
−

8. Try This
Evaluate
without
using a
calculator:
1
29 9 3= ±
2
38
−
3 2
1 1
48
=

9. Try This
Evaluate:
1 1
2 2
64 64
−
−
7 63
7
8 8
=

10. Example
Express using rational
exponents:
3 2
x
2 4 84
16a b c

11. Try This
Express using rational
exponents:
5 3
a
3
5
a

12. Try This
Express using rational
exponents:
3 2 63
8x y z
2
23
2xy z

13. Example
Express using radicals:
2
5
x
2 1
3 3
a b
11 3
82 4
16 x y ( )
11
10 2 102x y a

14. Try This
Express using radicals:
3 205
15x y
1 3
45 5
15 x y

15. Try This
Express using radicals:
6 10
16s t
3 5
4s t

16. Definition
Perfect Squares:the number
or expression resulting
from multiplying any whole
number or expression by
itself.

17. Examples
perfect square because:
•1
•4
•9
•16
1 x 1=1
2 x 2=4
3 x 3 =9
4 x 4=16

18. Try This
On a sheet of paper,list all
the perfect squares from
16 to 625.

19. Solution
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625

20. Example
perfect square because:
2
x
4
x
6
y
8
( )xy
2
x x x=g
2 2 4
x x x=g
3 3 6
y y y=g
4 4 8
( ) ( ) ( )xy xy xy=g
Do you
see a
pattern?

21. Example
Simplify
by
removing
all perfect
square
factors:
8
12
2
y
5
200xy

22. Try This
Simplify
by
removing
all perfect
square
factors:
200
10 2

23. Try This
Simplify
by
removing
all perfect
square
factors:
48
4 3

24. Try This
Simplify
by
removing
all perfect
square
factors:
28
2 7

25. Try This
Simplify
by
removing
all perfect
square
factors:
63
3 7

26. Try This
Simplify by
removing
all perfect
square
factors:
5 2 5 7
a b c
5 2 2
bc a c

27. Try This
Simplify
by
removing
all perfect
square
factors:
5
12x
2
2 3x x

28. Try This
Simplify
by
removing
all perfect
square
factors:
4 3
x y
2
x y y

29. Definition
Radical multiplication:
a b ab• =
ab a b= •
and

30. Example
2 3 =g
6 =
x y z =
6 x =

31. Try This
2
6 2x =g 2
12x
Can you write this answer
in simpler form...

32. Definition
Radical Division:
a a
bb
=
a a
b b
=
and

33. Example
6
3
=
12 =

34. Definition
Radical Addition &
Subtraction:
( )a x b x a b x+ = +
( )a x b x a b x− = −
( )a x b y a b xy+ ≠ +

35. Example
2 x x+ =
5 7 3 7− =
7 3p q+ =

36. Important Idea
x
The radicand
In order to
add or
subtract
radicals, the
radicands
must be the
same

37. Try This
5 3xy xy+ = 8 xy

38. Try This
5 3xy ab+ = 5 3xy ab+
The radicands are not the
same, therefore the terms
cannot be combined.

39. Example
6 54+
Add:
2 2 3 8+

40. Important Idea
Sometimes you can add
unlike radicands after
removing perfect square
factors

41. Try This
Add:
2 3 48−
2 3−

42. Definition
Rationalizing the
Denominator:
A procedure for writing an
equivalent expression
without any radicals in the
denominator.

43. Example
Rationalize the Denominator:
1
3

44. Example
Rationalize the Denominator:
3
5

45. Try This
Rationalize the Denominator:
2 11
5
2 55
5

46. Example
Rationalize the Denominator:
2
3
x
y

47. Try This
Rationalize the Denominator:
3 2
2
x
y
3 2
2
xy
y

48. Example
Rationalize the Denominator:
2
2
x
y+

49. Try This
Rationalize the Denominator:
x y
x y
+
−
( )
2
x y
x y
+
−
2x xy y
x y
+ +
−
or

50. Definition
Simplified Radical Form:
•No perfect square factors
in the radicand
•No fractional radicands
•No radicals in denominator

51. Important Idea
On tests and homework,
you are expected to
leave answers in
simplified radical form
where appropriate.

52. Try This
Re-write using simplified
radical form:
1
4
1
2

53. Try This
Re-write using simplified
radical form:
1
4
1
2

54. Try This
Re-write using simplified
radical form:
3
12x 2 3x x

55. Try This
Re-write using simplified
radical form:
4 2
xy
−
4 2
14
xy xy+

56. Properties of Exponents
Product Property
m n m n
a a a +
=

57. Properties of Exponents
Power of a Power Property
( )m n mn
a a=

58. Properties of Exponents
Power of a Quotient Property
, 0
m m
m
a a
b
b b
 
= ≠ ÷
 

59. Properties of Exponents
Power of a Product Property
( )m m m
ab a b=

60. Properties of Exponents
Quotient Property
, 0
m
m n
n
a
a a
a
−
= ≠

61. Properties of Exponents
Identity Property
1
a a=

62. Properties of Exponents
Reciprocal Property
1m
m
a
a
−
=
Do not leave answers
with negative exponents

63. Properties of Exponents
0
1a =

64. Try This
Evaluate
3 2
6 6
3 2 5
6 6+
=
1 1
3
3
−
=
2
3
3
3

65. Try This
Evaluate
( )
32
2 2 3
2 64x
=
2 4 4
2 4x x=( )
22
2x
( )
04 2
13 3x x− − 1

66. Example
Simplify
using only
positive
exponents:
2 2
((2 ) )x −

67. Try This
Simplify
using only
positive
exponents:
2 2 3
(2 )x y−
6
6
8x
y

68. Example
2
3 3
348r s−
 
 ÷
 ÷
 
Simplify using only
positive exponents:

69. Example
Simplify using only
positive exponents:
1 3 3
2 4 2x x x
 
− ÷
 ÷
 

70. Example
Simplify using only
positive exponents:
25 7
42 4x y xy
−
  
 ÷ ÷
 ÷ ÷
  

71. Try This
Simplify using only positive
exponents: 5
2 4
4516u v−
 
 ÷
 ÷
 1
2
5
32u
v
Can you think of
another way to
write the answer?

72. Try This
Simplify using only positive
exponents:
Can you think of
another way to
write the answer?
27 5
33 6x y xy
−
  
 ÷ ÷
 ÷ ÷
  
1 4
3 3x y

73. Lesson Close
Radical and exponential
expressions are important
for the mathematical
description of a variety of
problems in the physical
sciences and business.

74. Practice
1. pws5.1a (slides 1-15)
2. pws5.1b prob. 1-15 (slides 16-41)
3. pws5.1b prob. 16-31 (slides 42-55)
4. 334/55,56,63-72 all (slides 56-74)