ELEMENTARY ALGEBRA
MAT 0024
Prof. Howard Sorkin
SUPPLEMENTARY SHEET 5
NEGATIVE EXPONENTS and SCIENTIFIC NOTATION

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GO TO SCIENTIFIC NOTATION 
 
NEGATIVE EXPONENTS 

The basic concept of the NEGATIVE EXPONENT is to take the INVERSE or RECIPROCAL of the base to the POSITIVE power.

  
  
Rule 1:
CLICK HERE  for explanation of Rule 1
  Proof:
  By the DIVISION RULE:
  Therefore:
   Examples:  
   
CLICK HERE  for explanation of Examples a, b, c and d
   
a)
 
  
  
b)      
 
  
   
c)      
 
  
 
d)  REMEMBER TO USE ORDER OF OPERATIONS:   
   
 

   
FIRST the EXPONENT.....THEN the DOUBLE NEGATIVE

 
  
  
CLICK HERE  for explanation of Example e and NEW QUICK and EASY WAY to add or
subtract two fractions that do not have a common factor other than 1 in their denominators.

e)  FIRST write WITHOUT the NEGATIVE EXPONENT...  
   
 
   
                       ...THEN do the ADDITION PROBLEM 
                       This problem was done by first finding the 
                       least common denominator. 

We will learn a NEW quick and easy way to add or subtract two fractions that do not have a common factor other than 1 in their denominators:

 

Using the NEW quick and easy way to add the two fractions we have:
   

 
 
 
   

 
  
Rule 2:
 
CLICK HERE  for explanation of Rule 2
  
Proof: (using Rule 1)
This is really a DIVISION problem
 
   Examples:  
     
CLICK HERE  for explanation of Examples a, b, c and d

a)
 
  
  
b)  
 
  
  
c)  REMEMBER TO USE ORDER OF OPERATIONS:   
   
 
   
FIRST the EXPONENT.....THEN the DOUBLE NEGATIVE
  
  
d)
  
Remember: A negative exponent means the INVERSE or RECIPROCAL of the
BASE to the POSITIVE power.   In this example  is the BASE.  
   
What we are doing is taking the INVERSE or RECIPROCAL of the BASE to the
POSITIVE power.
 
 
 
 
 
 
GO TO PROBLEMS ON NEGATIVE EXPONENTS 
 
 
SCIENTIFIC NOTATION - 
MULTIPLICATION and DIVISION  
 
CLICK HERE  for explanation of MULTIPLYING using EXPONENTIAL NOTATION

Suppose we want to find the answer to the following multiplication problem, written in
STANDARD FORM. 

Since we know multiplication is COMMUTATIVE, we can rewrite this  problem as
follows:

Of course, there really is no need to rewrite this problem if we are able to calculate
  and  in our heads. 
CLICK HERE  for explanation of DIVIDING using EXPONENTIAL NOTATION
 

We will now find the answer, in STANDARD FORM, to the following  division problem: 

Since we want our answer to be in STANDARD FORM, when we DIVIDE we
ALWAYS bring the  number UP and work out the answer. 

REMEMBER, when we move a number either UP or DOWN, the sign of  the
exponent changes (see RULE 1 and RULE 2 above). 

Here are the steps involved with finding the answer to the DIVISION 
problem above. 
  
1.  Simplify the  part of the problem. 

2.  Bring  UP (it now becomes ) and find this answer. 

3.  Write the final answer in STANDARD FORM. 
    

Problems 49 - 56 below deal with MULTIPLICATION and DIVISION using
SCIENTIFIC NOTATION.

 
  Find the answers to the following problems:
  1.      2.    3.    4. 
  5.      6.     7.     8.  
  9.  
10.  Solutions:
1 - 10
11.   12.  
13.    14.   15.   16. 
17.    18   19.  
20.  Solutions:
11 - 20
21.   22.   23.   24.  
25.   26.   27.   28. 
29. 
30.  Solutions: 21 - 30
31.  32. 
33.  34. 
35. Solutions: 31 - 35
36. 
37.   38.   39.  
40. Solutions: 36 - 40
41.  
 
 42. 
 
43.   
 
44.  
45.  
 
46.  
 
47.  
 
  48. 
49.  
50.  
51.  
52. 
 
 
  		 

 

J J J J J J J J J J J J J J J
ANSWERS

J J J J J J J J J J J J J J J
1.  2.  3.  4.  5.  6.  7.  8. 
9.   3 10.  -3 11. -3 12.  3 13.  14.  15.  16. 
17. 1 18.  -1 19.  1 20.  2 21.  1  22.  -2  23.  1 24. 3x2y
25.  -4xy  26.  -4x  27.  -4 28.  0 29.  -2  30.  0  31.  -2  32. 
33.   34.   35.  36.   37.   38.  39.   40. 
41.   42.   43.  44.  45.  46.  47.  48. 
49.   584,000,000 
50.   .078 
51.   1.596 
52.   .000168		 53.   260 
54.   .000027 
55.   850,000 
56.   .0005

Back to  NEGATIVE EXPONENT PROBLEMS 
Back to  SCIENTIFIC NOTATION PROBLEMS
© Howard Sorkin 1986, 1998, 2000, 2004   All rights reserved