- Powers - Roots - Sequences

2. edwinxav@hotmail.com
elapuerta@hotmail.com

3. CONTENTS

4. POWERS

6. 1
2
3
... n
n times
a a
a a a
a a a a
a a a a a

 
  
    
EXPONENTS
Square
Cube

8. 0 a 1, when a  0
ZERO EXPONENT
0
0
0
2 1
5 1
1
1
4


 
  
 

9. when:

 , and n is even
0 n a 
a  0
a  0
PROPERTIES OF THE
EXPONENTS
 
 
 
4
3
4
2 16
2 8
2 16


 

10. when:
 , and n is odd
0 n a 
a  0
PROPERTIES OF THE
EXPONENTS
 
 
3
3
2 8
3 27
  
  

11. even
odd
positive positive
positive positive


even
odd
negative positive
negative negative


ODD/EVEN EXPONENTS
4
5
2 16
2 32


 
 
4
5
2 16
2 32
 
  

12.  , when n > 0
 is undefined
0n 
0 0 n 
0 0
POWERS OF ZERO

13. 2 m m m a  a  a
ADDITION OF POWERS
3 3 3 a  a  2a
2 2 2 3a  5a  8a

14.  m m m a  b  a  b
If a  0, b  0, and m  1, then
 2 2 2 a  b  a  b

15. 0 m m a  a 
SUBTRACTION OF
POWERS
3 3 a  a  0
2 2 2 7a  4a  3a

16.  m m m a b  a b
If a  0, b  0, a  b, and m  1, then
 3 3 3 a b  a b

17. m n m n a a a   
MULTIPLICATION OF
POWERS
23 24  27
    7 2 2 2  2 2 2 2  2

18. 7
4
3
2
2
2

DIVISION OF POWERS
m
m n
n
a
a
a
 
4 2 2 2 2 2 2 2
2 2 2 2 2
2 2 2
     
    
 

19. k 1 k a a a    1
k
k a
a
a
 
MULTIPLICATION/DIVISION
OF POWERS

20.  n
m m n a a 
POWERS TO A POWER
 4
23  212
          4
12
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2
             


21.  m m m ab  a b
POWER OF A PRODUCTS
 2 2 2 23  2 3
      2
2 2
2 3 2 3 2 3
2 3
    
 

22.  p
m n mp np a b  a b
 2
4 5 8 10 2 3  2 3
      2
4 5 4 5 4 5
4 5 4 5
8 10
2 3 2 3 2 3
2 3 2 3
2 3
    
   
 

23. 2 2
2
3 3
4 4
 
  
 
POWER OF QUOTIENTS
m m
m
a a
b b
 
  
 
2 2
2
3 3 3 3
4 4 4 4
     
      
     

24. n a 
1 1
n
n
n a
a a
  
  
 
1 n
n a
a 
NEGATIVE EXPONENTS
3
3
1
2
2
 
3
3
1
2
2 

25. NEGATIVE EXPONENTS
1nnaa 

26. POWERS

27. Edwin Lapuerta, May 2014
ROOTS

29. 3
... n
n times
b a a a b
c a a a a c
d a a a a a d
   
    
      
ROOTS
Square root
Cubic root

31. a b a b
a b a b
  
  
ADDITION/SUSTRACTION
OF ROOTS
9 16 9 16
25 16 25 16
  
  

32. a b ab
ab a b
 
 
MULTIPLYING ROOTS
4 9 36 6
36 4 9 4 9 6
  
    

33. a a
b b
a a
b b


DIVIDING ROOTS
36 36
2
9 9
36 36
2
9 9
 
 

34. a a b
b b b
a b
b
 

RATIONALIZATION
2 2 3
3 3 3
2 3
3
 


35. 1
m
n n m
n n
a a
a a


FRACTIONAL EXPONENT
2
3 3 2
1
2 2 1
8 8 4
9 9 9 3
 
  

36.  
1
1
n
n
n n
a a
a a
 
  
 

If a ≥ 0
FRACTIONAL EXPONENT
 
3
1
3
1
3 3
2 2
2 2
 
  
 


37.  n
n
n n
a a
a a


If a ≥ 0
FRACTIONAL EXPONENT
 3
3
3 3
2 2
2 2



38. m n m n
m n n m
a a
a a


If a ≥ 0
ROOT OF A ROOT
2 3 2 3 6
2 3 3 2 6
64 64 64 2
64 64 64 2
  
  

39. ROOTS

40. SOLVING EXPONENTIAL EQUATIONS
If ax= ay, then x= y( a≠ 0 and a≠ 1).

41. EXPONENTIAL EQUATIONS

42. SEQUENCES

44. SEQUENCE
The first term of a sequence is represented by a1, the second term a2, and so on to the nth term, an.

45. 2 1 1 2 ..., , , , , ,... n n n n n a a a a a    
SEQUENCE
..., a2 , a3, a4 , a5 , a6 , ...

46. ARITHMETIC SEQUENCES
A sequence in which each term, after the first, if found by adding a constant, called the common difference, to the previous term.
2, 5, 8, 11, 14, …

47. 2
5
8
11
14
a1
a2
a3
a4
a5
2, 5, 8, 11, 14, …

48.   1 1 1 1 1
1 2 3 4
, , 2 , 3 ,..., 1
, , , ,..., n
a a d a d a d a n d
a a a a a
    
2, 5, 8, 11, 14, …
an  a1 n 1d

51. GEOMETRIC SEQUENCES
A sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio.
2, 6, 18, 54, 162, …

52. a1
a2
a3
a4
a5
2, 6, 18, 54, 162, …

53. 2 3 1
1 1 1 1 1
1 3 3 4
, , , ,...,
, , , ,...,
n
n
a a r a r a r a r
a a a a a
    
2, 6, 18, 54, 162, …
1
1
n
n a a r   

54. Apple Inc. Cash Growth
BullishCross 2013 Outlook
(in millions)

55. ARITHMETIC AND GEOMETRIC SEQUENCES

58. SEQUENCES

59. SUMMARY

60. edwinxav@hotmail.com
elapuerta@hotmail.com