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21

NEGATIVE EXPONENTS

Power of a fraction

Section 2

Power of a fraction

 

"To raise a fraction to a power, raise the numerator
and denominator to that power."

   Example 1.  

For,  according to the meaning of the exponent, and the rule for multiplying fractions:

   Example 2.   Apply the rules of exponents:   

Solution.   We must take the 4th power of everything.  But to take a power of a power -- multiply the exponents:

Problem 1.   Apply the rules of exponents.

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  a)   = x2
y2
  b)   = 8x3
27
  c)   =
  d)   =
  e)   = x2 − 2x + 1
x2 + 2x + 1
   Perfect Square Trinomial

Subtracting exponents

In the previous Lesson we saw the following rule for reducing a fraction:

ax
ay
= x
y

"Both the numerator and denominator may be divided
 by a common factor."

Consider these examples:

2· 2· 2· 2· 2
     2· 2
= 2· 2· 2
___2· 2___
2· 2· 2· 2· 2
= __1__
2· 2· 2

If we write those with exponents, then


22
= 23
22
25
=  1 
23

In each case, we subtract the exponents.  But when the exponent in the denominator is larger, we write 1-over the difference.

  Example 3.   
x3
= x5
 
x8
=  1 
x5

Here is the rule:

Problem 2.   Simplify the following.  (Do not write a negative exponent.)

  a)   = x3   b)   x2
x5
=  1 
x3
  c)    x
x5
=  1
x4
  d)   x2
x
= x   e)   = x4   f)   =  1
x2

Problem 3.   Simplify each of the following.  Then calculate each number.

  a)   = 23 = 8   b)   22
25
=  1 
23
= 1
8
  c)    2
25
=  1
24
=  1 
16
  d)   22
2
= 2   e)   = −24 = −16.  See Lesson 13.
  f)   =  1
22
= 1
4
   Example 4.   Simplify by reducing to lowest terms:   

Solution.   Consider each element in turn:

Problem 4.   Simplify by reducing to lowest terms.  (Do not write negative exponents.

  a)   =   y3
5x3
  b)    = 8a3
5b3
  c)   =   3z_
5x4y3
  d)    =   c3
16
 e)   (x + 1)3 (x − 1)
(x − 1)3 (x + 1)
= (x + 1)2
(x − 1)2

Negative exponents

We are now going to extend the meaning of an exponent to more than just a positive integer.  We will do that in such a way that the usual rules of exponents will hold.  That is, we will want the following rules to hold for any exponents:  positive, negative, 0 -- even fractions.

aman = am + n   Same Base
 
(ab)n = anbn   Power of a Product
 
(am)n = amn   Power of a Power

We begin by defining a number with a negative exponent.

an  =    1 
an

It is the reciprocal of that number with a positive exponent.

an is the reciprocal of an.

  Example 5.     2−3 =  1
23
= 1
8

The base, 2, does not change.  The negative exponent becomes positive -- in the denominator.

Example 6.   Compare the following. That is, evaluate each one:

3−2    −3−2   (−3)−2    (−3)−3

  Answers.  3−2  =   1
32
 =  1
9
.

Next,

−3−2 is the negative of 3−2. (See Lesson 13.)  The base is still 3.

−3−2  = − 1
9
.

As for (−3)−2, the parentheses indicate that the base is −3:

(−3)−2  =     1   
(−3)2
 =  1
9
.

Finally,

(−3)−3  =     1   
(−3)3
 =   1 
27
.

A negative exponent, then, does not produce a negative number.  Only a negative base can do that. And then the exponent must be odd

  Example 7.   Simplify   a2
a5
.

Solution.   Since we have invented negative exponents, we can now subtract any exponents as follows:

a2
a5
= a2 − 5 = a−3

We now have the following rule for any exponents m, n:

In fact, it was because we wanted that rule to hold that we

  defined an as    1 
an
.

We want

a2
a5
= a−3

But

a2
a5
=  1 
a3
Therefore, we define  a−3 as   1 
a3
.
  Example 8.    a−1 = 1
a

a−1 is now a symbol for the reciprocal, or multiplicative inverse, of any number a.  It appears in the following rule (Lesson 5):

a· a−1 = 1

Problem 5.   Evaluate the following.

   a)  ( 2
3
)−1   =   3
2
.   3
2
 is the reciprocal of  2
3
.
   b)  ( 2
3
)−4   =   81
16
.   81
16
 is the reciprocal of the 4th power of   2
3
.
   Problem 6.   Show:  am· bn  =   am
bn
.
am· bn  =  am·  1
bn
  =   am
bn
.    The definition of division.

Example 9.   Use the rules of exponents to evaluate  (2−3· 104)−2.

 Solution.  (2−3· 104)−2 = 26· 10−8  Power of a power
 
    =  Problem 6
 
    =       _64_      
100,000,000

Problem 7.   Evaluate the following.

  a)   2−4  =   1 
24
 =   1
16
    b)   5−2  =   1 
52
 =   1
25
    c)   10−1  =   1 
101
 =   1
10
  d)   (−2)−3  =     1   
(−2)3
 =    1  
−8
 =  1
8
  e)   (−2)−4  =     1   
(−2)4
 =   1 
16
  f)   −2−4  =   1 
24
 =   1 
16

g)  (½)−1 =  2.   2 is the reciprocal of ½.

Problem 8.   Use the rules of exponents to evaluate the following.

   a)   102· 10−4 = 102 − 4 = 10−2 = 1/100.
   b)   (2−3)2   =   2−6   =    1 
26
  =    1 
64
   c)   (3−2· 24)−2   =   34· 2−8   =   34
28
  =    81 
256
   d)   2−2· 2   =   2−2+1   =   2−1   =   1
2

Problem 9.   Rewrite without a denominator.

  a)   x2
x5
= x2−5 = x−3   b)    y
y6
= y1−6 = y−5
  c)   = x−3y−4   d)   = a−1b−6c−7
  e)   1
x
= x−1   f)    1
x3
= x−3
 g)   (x + 1)
    x
 =  (x + 1)x−1     h)   (x + 2)2
(x + 2)6
= (x + 2)−4

Example 10.     Rewrite without a denominator, and evaluate:

Answer.   The rule for subtracting exponents --

-- holds even when an exponent is negative.

Therefore,

= 10−3 + 5 − 2 + 4 = 104 = 10,000.

Exponent 2 goes into the numerator as −2;  exponent −4 goes there as +4.

Problem 10.    Rewrite without a denominator and evaluate.

 a)    22
2−3
22 + 3  = 25 = 32   b)    102
10−2
102 + 2  = 104 = 10,000
 c)   102 − 5 − 4 + 6  = 10−1 =  1 
10
 d)   25 − 6 + 9 − 7  = 21 = 2

The reciprocal of an.

Reciprocals come in pairs.  The reciprocal of  an  is  an :

 1 
an
 =   an .

And the reciprocal of  an  is  an :

  1  
an
 =   an .

That implies:

Factors may be shifted between the denominator and the numerator
by changing the sign of the exponent.

  Example 11.   Rewrite without a denominator:  
  Answer.     

The exponent 3 goes into the numerator as −3;  the exponent −4 goes there as +4.

Problem 11.    Rewrite with positive exponents only.

 a)    x 
y−2
= xy2   b)   =   c)   =
d)   =   e)   =

Problem 12.    Apply the rules of exponents, then rewrite with positve exponents.

a)   = =   b)   = =

Section 2: Exponent 0

Next Lesson:  Multiplying and dividing algebraic fractions


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