5-1: Radicals & Rational
Exponents
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define and apply rational
expone...
Definition
A rational number is any
number that can be written
in the form:
, 0
m
n
n
≠
Definition
A rational exponent is any
number that can be written
in the form:
, 0
m
n
b n ≠
Definition
m
n mn
b b=
Rational exponent form
Radical form
Definition
m
n
b b
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...
Example
Evaluate without using a
calculator:
1
2
81
2
3
27
−
Try This
Evaluate
without
using a
calculator:
1
29 9 3= ±
2
38
−
3 2
1 1
48
=
Try This
Evaluate:
1 1
2 2
64 64
−
−
7 63
7
8 8
=
Example
Express using rational
exponents:
3 2
x
2 4 84
16a b c
Try This
Express using rational
exponents:
5 3
a
3
5
a
Try This
Express using rational
exponents:
3 2 63
8x y z
2
23
2xy z
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
Try This
Express using radicals:
3 205
15x y
1 3
45 5
15 x y
Try This
Express using radicals:
6 10
16s t
3 5
4s t
Definition
Perfect Squares:the number
or expression resulting
from multiplying any whole
number or expression by
itself.
Examples
perfect square because:
•1
•4
•9
•16
1 x 1=1
2 x 2=4
3 x 3 =9
4 x 4=16
Try This
On a sheet of paper,list all
the perfect squares from
16 to 625.
Solution
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
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 y...
Example
Simplify
by
removing
all perfect
square
factors:
8
12
2
y
5
200xy
Try This
Simplify
by
removing
all perfect
square
factors:
200
10 2
Try This
Simplify
by
removing
all perfect
square
factors:
48
4 3
Try This
Simplify
by
removing
all perfect
square
factors:
28
2 7
Try This
Simplify
by
removing
all perfect
square
factors:
63
3 7
Try This
Simplify by
removing
all perfect
square
factors:
5 2 5 7
a b c
5 2 2
bc a c
Try This
Simplify
by
removing
all perfect
square
factors:
5
12x
2
2 3x x
Try This
Simplify
by
removing
all perfect
square
factors:
4 3
x y
2
x y y
Definition
Radical multiplication:
a b ab• =
ab a b= •
and
Example
2 3 =g
6 =
x y z =
6 x =
Try This
2
6 2x =g 2
12x
Can you write this answer
in simpler form...
Definition
Radical Division:
a a
bb
=
a a
b b
=
and
Example
6
3
=
12 =
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+ ≠ +
Example
2 x x+ =
5 7 3 7− =
7 3p q+ =
Important Idea
x
The radicand
In order to
add or
subtract
radicals, the
radicands
must be the
same
Try This
5 3xy xy+ = 8 xy
Try This
5 3xy ab+ = 5 3xy ab+
The radicands are not the
same, therefore the terms
cannot be combined.
Example
6 54+
Add:
2 2 3 8+
Important Idea
Sometimes you can add
unlike radicands after
removing perfect square
factors
Try This
Add:
2 3 48−
2 3−
Definition
Rationalizing the
Denominator:
A procedure for writing an
equivalent expression
without any radicals in the
den...
Example
Rationalize the Denominator:
1
3
Example
Rationalize the Denominator:
3
5
Try This
Rationalize the Denominator:
2 11
5
2 55
5
Example
Rationalize the Denominator:
2
3
x
y
Try This
Rationalize the Denominator:
3 2
2
x
y
3 2
2
xy
y
Example
Rationalize the Denominator:
2
2
x
y+
Try This
Rationalize the Denominator:
x y
x y
+
−
( )
2
x y
x y
+
−
2x xy y
x y
+ +
−
or
Definition
Simplified Radical Form:
•No perfect square factors
in the radicand
•No fractional radicands
•No radicals in de...
Important Idea
On tests and homework,
you are expected to
leave answers in
simplified radical form
where appropriate.
Try This
Re-write using simplified
radical form:
1
4
1
2
Try This
Re-write using simplified
radical form:
1
4
1
2
Try This
Re-write using simplified
radical form:
3
12x 2 3x x
Try This
Re-write using simplified
radical form:
4 2
xy
−
4 2
14
xy xy+
Properties of Exponents
Product Property
m n m n
a a a +
=
Properties of Exponents
Power of a Power Property
( )m n mn
a a=
Properties of Exponents
Power of a Quotient Property
, 0
m m
m
a a
b
b b
 
= ≠ ÷
 
Properties of Exponents
Power of a Product Property
( )m m m
ab a b=
Properties of Exponents
Quotient Property
, 0
m
m n
n
a
a a
a
−
= ≠
Properties of Exponents
Identity Property
1
a a=
Properties of Exponents
Reciprocal Property
1m
m
a
a
−
=
Do not leave answers
with negative exponents
Properties of Exponents
0
1a =
Try This
Evaluate
3 2
6 6
3 2 5
6 6+
=
1 1
3
3
−
=
2
3
3
3
Try This
Evaluate
( )
32
2 2 3
2 64x
=
2 4 4
2 4x x=( )
22
2x
( )
04 2
13 3x x− − 1
Example
Simplify
using only
positive
exponents:
2 2
((2 ) )x −
Try This
Simplify
using only
positive
exponents:
2 2 3
(2 )x y−
6
6
8x
y
Example
2
3 3
348r s−
 
 ÷
 ÷
 
Simplify using only
positive exponents:
Example
Simplify using only
positive exponents:
1 3 3
2 4 2x x x
 
− ÷
 ÷
 
Example
Simplify using only
positive exponents:
25 7
42 4x y xy
−
  
 ÷ ÷
 ÷ ÷
  
Try This
Simplify using only positive
exponents: 5
2 4
4516u v−
 
 ÷
 ÷
 1
2
5
32u
v
Can you think of
another way to...
Try This
Simplify using only positive
exponents:
Can you think of
another way to
write the answer?
27 5
33 6x y xy
−
  ...
Lesson Close
Radical and exponential
expressions are important
for the mathematical
description of a variety of
problems i...
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,...
Upcoming SlideShare
Loading in …5
×

Hprec5.1

945 views
651 views

Published on

Published in: Technology, Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
945
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
9
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Hprec5.1

  1. 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. 2. Definition A rational number is any number that can be written in the form: , 0 m n n ≠
  3. 3. Definition A rational exponent is any number that can be written in the form: , 0 m n b n ≠
  4. 4. Definition m n mn b b= Rational exponent form Radical form
  5. 5. Definition m n b b
  6. 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. 7. Example Evaluate without using a calculator: 1 2 81 2 3 27 −
  8. 8. Try This Evaluate without using a calculator: 1 29 9 3= ± 2 38 − 3 2 1 1 48 =
  9. 9. Try This Evaluate: 1 1 2 2 64 64 − − 7 63 7 8 8 =
  10. 10. Example Express using rational exponents: 3 2 x 2 4 84 16a b c
  11. 11. Try This Express using rational exponents: 5 3 a 3 5 a
  12. 12. Try This Express using rational exponents: 3 2 63 8x y z 2 23 2xy z
  13. 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. 14. Try This Express using radicals: 3 205 15x y 1 3 45 5 15 x y
  15. 15. Try This Express using radicals: 6 10 16s t 3 5 4s t
  16. 16. Definition Perfect Squares:the number or expression resulting from multiplying any whole number or expression by itself.
  17. 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. 18. Try This On a sheet of paper,list all the perfect squares from 16 to 625.
  19. 19. Solution 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625
  20. 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. 21. Example Simplify by removing all perfect square factors: 8 12 2 y 5 200xy
  22. 22. Try This Simplify by removing all perfect square factors: 200 10 2
  23. 23. Try This Simplify by removing all perfect square factors: 48 4 3
  24. 24. Try This Simplify by removing all perfect square factors: 28 2 7
  25. 25. Try This Simplify by removing all perfect square factors: 63 3 7
  26. 26. Try This Simplify by removing all perfect square factors: 5 2 5 7 a b c 5 2 2 bc a c
  27. 27. Try This Simplify by removing all perfect square factors: 5 12x 2 2 3x x
  28. 28. Try This Simplify by removing all perfect square factors: 4 3 x y 2 x y y
  29. 29. Definition Radical multiplication: a b ab• = ab a b= • and
  30. 30. Example 2 3 =g 6 = x y z = 6 x =
  31. 31. Try This 2 6 2x =g 2 12x Can you write this answer in simpler form...
  32. 32. Definition Radical Division: a a bb = a a b b = and
  33. 33. Example 6 3 = 12 =
  34. 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. 35. Example 2 x x+ = 5 7 3 7− = 7 3p q+ =
  36. 36. Important Idea x The radicand In order to add or subtract radicals, the radicands must be the same
  37. 37. Try This 5 3xy xy+ = 8 xy
  38. 38. Try This 5 3xy ab+ = 5 3xy ab+ The radicands are not the same, therefore the terms cannot be combined.
  39. 39. Example 6 54+ Add: 2 2 3 8+
  40. 40. Important Idea Sometimes you can add unlike radicands after removing perfect square factors
  41. 41. Try This Add: 2 3 48− 2 3−
  42. 42. Definition Rationalizing the Denominator: A procedure for writing an equivalent expression without any radicals in the denominator.
  43. 43. Example Rationalize the Denominator: 1 3
  44. 44. Example Rationalize the Denominator: 3 5
  45. 45. Try This Rationalize the Denominator: 2 11 5 2 55 5
  46. 46. Example Rationalize the Denominator: 2 3 x y
  47. 47. Try This Rationalize the Denominator: 3 2 2 x y 3 2 2 xy y
  48. 48. Example Rationalize the Denominator: 2 2 x y+
  49. 49. Try This Rationalize the Denominator: x y x y + − ( ) 2 x y x y + − 2x xy y x y + + − or
  50. 50. Definition Simplified Radical Form: •No perfect square factors in the radicand •No fractional radicands •No radicals in denominator
  51. 51. Important Idea On tests and homework, you are expected to leave answers in simplified radical form where appropriate.
  52. 52. Try This Re-write using simplified radical form: 1 4 1 2
  53. 53. Try This Re-write using simplified radical form: 1 4 1 2
  54. 54. Try This Re-write using simplified radical form: 3 12x 2 3x x
  55. 55. Try This Re-write using simplified radical form: 4 2 xy − 4 2 14 xy xy+
  56. 56. Properties of Exponents Product Property m n m n a a a + =
  57. 57. Properties of Exponents Power of a Power Property ( )m n mn a a=
  58. 58. Properties of Exponents Power of a Quotient Property , 0 m m m a a b b b   = ≠ ÷  
  59. 59. Properties of Exponents Power of a Product Property ( )m m m ab a b=
  60. 60. Properties of Exponents Quotient Property , 0 m m n n a a a a − = ≠
  61. 61. Properties of Exponents Identity Property 1 a a=
  62. 62. Properties of Exponents Reciprocal Property 1m m a a − = Do not leave answers with negative exponents
  63. 63. Properties of Exponents 0 1a =
  64. 64. Try This Evaluate 3 2 6 6 3 2 5 6 6+ = 1 1 3 3 − = 2 3 3 3
  65. 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. 66. Example Simplify using only positive exponents: 2 2 ((2 ) )x −
  67. 67. Try This Simplify using only positive exponents: 2 2 3 (2 )x y− 6 6 8x y
  68. 68. Example 2 3 3 348r s−    ÷  ÷   Simplify using only positive exponents:
  69. 69. Example Simplify using only positive exponents: 1 3 3 2 4 2x x x   − ÷  ÷  
  70. 70. Example Simplify using only positive exponents: 25 7 42 4x y xy −     ÷ ÷  ÷ ÷   
  71. 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. 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. 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. 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)

×