Transcript Lesson 6.5: Looking Back with Exponents

•
To review or learn the division property of exponents
• In the previous lesson you learned that
looking ahead in time to predict future
growth with an exponential model is
related to the multiplication property of
exponents.
• In this lesson you’ll discover a rule for
dividing expressions with exponents.
Then you’ll see how dividing expressions
is like looking back in time.
The Division Property of Exponents
• Step 1:
▫ Write the numerator and the denominator of each
quotient in expanded form.
▫ Then reduce to eliminate common factors.
▫ Rewrite the factors that remain with exponents.
Use your calculator to check your answers.
9
5
56
3
3
3 5
3  52
44 x 6
42 x 3
9
5
56
1
1
1
1
1
1
1
1
5555555555 5  5  5  5  5  5 555


 53
555555
5 5 5 5 5 5
1
1
1
1
1
1
1
33  53
3  3  3  5  5  5 3  3  3  5  5  5 32  5
2




3
5
2
35
355
1
3 5 5
1
44 x 6
42 x 3

1
1
4444 x  x  x  x  x  x
44 x  x  x
1
1
1
1
1
4 4 4  4  x  x  x  x  x  x

4 4 x  x  x
1
1
42 x 3

 42 x 3
1
1
1
1
• Step 2:
▫ Compare the exponents in each final expression
you got in Step 1 to the exponents in the original
quotient.
▫ Describe a way to find the exponents in the final
expression without using expanded form.
59
3

5
56
33  53
2

3
5
2
35
44 x 6
2 3

4
x
2 3
4 x
• Step 3:
▫ Use your method from Step 2 to rewrite
this expression so that it is not a fraction.
0.08
You can leave 12 as a fraction.

515  1 


511 1 

24
0.08 
12 
18
0.08 
12 

5 1 


11 
5 1 

15
24
0.08 
12 
18
0.08 
12 
24 18
15 11
5
0.08 

1  12 


0.08 

 5 1 
12 

4
6
• Recall that exponential growth is related to
repeated multiplication. When you look ahead
in time you multiply by repeated constant
multiplication, or increase the exponent.
• To look back in time you will need to undo some
of the constant multipliers, or divide.
• Step 4
▫ Apply what you have discovered about dividing
expressions with exponents.
 After 7 years the balance in a saving account is
500(1+0.04)7. What does the expression
500 1  0.04 
mean
in this situation?
3
1  0.04
7
 Rewrite this expression with a single exponent.
The balance 3 years prior.
500 1  0.04 
4
▫ After 9 years of depreciation,
the value of a car is
9
21,300 1-0.12 
,
1-0.12
5
21,300 1-0.12 
9
▫ What does the expression
mean
in this
5
1-0.12
situation?
▫ Rewrite this expression with a single exponent.
The balance 5 years prior
21,300 1-0.12 
4
▫ After 5 weeks the population of a bug colony is
32(1+0.50)5.
▫ Write a division expression to show the population
2 weeks earlier.
▫ Rewrite your expression with a single exponent.
32 1  0.50 
5
1  0.50
2
 32 1  0.50 
3
▫ The expression A(1+r)n can model n time periods
of exponential growth. What expression models
the growth m time periods earlier.
A 1  r 
n
1  r 
m
 A 1  r 
n-m
• Step 5
▫ How does looking back in time with an
exponential model relate to dividing expressions
with exponents?
Dividing by (1+r)m represents looking back m
time periods.
Expanded form helps you understand
many properties of exponents.
It also helps you understand how the
properties work together.
Example A
• Use the properties of exponents to rewrite each
expression.
6 x 9 6  x  x  x  x  x  x  x  x  x 6  x  x  x  x  x 6 x5



4
5 x  x  x  x
5
5
5x
3x 8x 
2
4
4 x 3
7.5 x108
1.5 x103
3  8  x2  x 4 3  8 x  x  x  x  x  x
3





6
x
4
xxx
4  x 3
7.5 x108 7.5 10  10  10  10  10  10  10  10
5




5.0
x10
10  10  10
1.5 x103 1.5
For any non-zero value of b and any
integer value of m and n
n
b
nm

b
m
b
Example B
• Six years ago, Anne bought a van for $18,500 for
her flower delivery service. Based on the prices
of similar used vans, she estimates a rate of
depreciation of 9% per year.
▫ How much is the van worth now?
A(1  r )x  18,500(1  0.09)6  10,505.58
▫ How much was the van worth last year?
18,500(1  0.09)5  11,544.59
▫ How much was it worth 2 years ago?
18,500(1  0.09)6 2  18,500(1  0.09)4  12,686.37