All About Alice
All About Alice
• In math, we love a good shortcut…
• If there are 20 students in a class and each student has $5,
how much do the students have in total?
5 + 5 + 5 + … + 5 = 5·20 = 100
…multiplication is just a shortcut for repeated addition.
• If Alice eats 7 ounces of cake, how many times bigger
will she grow?
2·2·2·2·2·2· = 27 = 128 times bigger
2
…exponents are a shortcut for repeated multiplication.
All About Alice
• The general form of an exponential function is
y  a bx
new amount
current amount
number of growth factors that
initial amount is multiplied by
(exponent)
growth factor
(base)
If Alice is 5 feet tall and eats 3 ounces of cake…
new height = 5·23 = 5·8 = 40 feet tall
If Alice is 12 feet tall and drinks 7 ounces of potion…
new height = 12·(½)5 = 12·0.03125 = 0.375 feet tall
All About Alice
• In A Wonderland Lost, we found that the area of the
forest, which is decreasing by 10% every year, could be
calculated using the exponential function
f ( y )  1,200,000  (0.90) y
because
f ( y )  1,200,000  (0.90)  (0.90)  ...  (0.90)
after 1 year
after 2 years
after y years
We used 0.90 as our base value because the new rainforest area is 100% –10% = 90% of the previous area.
All About Alice
• What if new laws and regulations were introduced so that
the area of the rainforest, currently 500,000 square miles,
will increase by 2% every year? What exponential
function could we use to calculate the area of the forest
after y years of growth?
f ( y )  500,000  (1.02)
y
The base value is 1.02 because the new rainforest area is
100% + 2% = 102% of the previous area.
Inflation, Depreciation, and Alice
All About Alice
• In general…
If the current amount is decreased by some %, then we
can find the base value of our exponential function by…
If the current amount is increased by some %, then we
can find the base value of our exponential function by…
Simplifying Expressions with Exponents
• Simplify
2

7
a
( x y) 
3
3
3 3
 4x

 
 y2 


5 3
3x y
6 y2

 a b 


 2 a  2b 


2 3
2

Simplifying Expressions with Exponents
• Additive Law of Exponents
When two exponential expressions with the same base
are multiplied, the following property holds:
bx · by
=
For example,
54 ·
53
bx+y
= (5·5·5·5)·(5·5·5 = 5·5·5·5·5·5· = 57
) 4
5
3
5
5
Simplifying Expressions with Exponents
• Additive Law of Exponents
When two exponential expressions with the same base
are multiplied, the following property holds:
bx · by
=
For example,
x7 ·
x5
bx+y
= (x·x·x·x·x·x·x)·(x·x·x·x·x
)
x7
x5
= x·x·x·x·x·x·x·x·x·x·x·x= x12
Simplifying Expressions with Exponents
• Simplify
h h 
3
5
( x 2 )(2 x5 ) 
(5 x y )( 2 xy ) 
2
3
2
7 a(a 2b)(  ab 4 c) 
negative sign
Simplifying Expressions with Exponents
• On your own, simplify each of the following expressions.
Then, check with a partner to see how you did.
(2a 5 )( 4a 2 ) 
( xy )(3x y ) 
5
2
4
(4a 2c 2 )( ab 4c)(  2b3c) 
Simplifying Expressions with Exponents
• Law of Repeated Exponentiation
When an exponential expression is raised to a power, the
following property holds:
y
x
(b )
=
bx·y
For example,
3
(x15) = (x15)·(x15)·(x15 = x15+15+15 = x15·3= x45
)
Simplifying Expressions with Exponents
• Simplify
(x ) 
6 3
(2 x 4 ) 2 
c (c ) 
5
2 7
( x ) (3xy ) 
2 2
2 4
Simplifying Expressions with Exponents
• On your own, simplify each of the following expressions.
Then, check with a partner to see how you did.
(5 y)
3
3 5 
2 4
(2a ) ( 3a )
7
5 8
5 y ( xy) (2 x )
Simplifying Expressions with Exponents
• Find the missing value that will make each equation true.
( x n )( x 3 )  ( x 9 )
a(a )  a
n 3
13
(b ) (b )  b
4 5
n 3
20
All About Alice
• Negative Exponents
• We have looked at many examples involving base values
and positive exponents. What happens if a base value
has a negative exponent?
23 =
22 =
21 =
20 =
2-1 =
2-2 =
2-3 =
All About Alice
• Negative Exponents
One way to think about this is that the negative exponent
just tells you that the base is on the wrong side of the
fraction line.
To take care of a negative exponent, just move the base
and (positive) exponent to the other side of the fraction
line.
x
2

1

3
b
2y
5

(10a) 4 
24( 2 x ) 4 
All About Alice
• Negative Exponents
When the base and exponent move, you must cancel out
any common factors in order for the expression to be
simplified.
5 x 2  x 8 
Does the additive law of exponents still work for
expressions with negative exponents?
All About Alice
• Negative Exponents
The law of repeated exponentiation (multiplying powers)
is also true for expressions with negative exponents.
5 2
(x ) 
(b 2 ) 3 
( y 4 ) 2 
All About Alice
• Negative Exponents
Simplify

2
7
3 1
(3a 4b 9 )
(8 x )( x )

(2 x 2 y 4 ) 3 (12 x 9 y 2 ) 
All About Alice
• Negative Exponents
There are also cases where the exponent expression is a
fraction with numbers and variables in both the
numerator and denominator
3
4x
x
4

9a
4 
6a
All About Alice
• Negative Exponents
Things can get complicated…
3
x y
2 4 
x y
8a 5  b 2

3
(2a)  b
3
x y
 4 5
 x y

6




2

All About Alice
• Negative Exponents
If the entire fraction is raised to a negative exponent, then
the numerator and denominator switch places, and the
negative exponent becomes positive.
x 
 3
y 
 
2
3
y
  2
x
3
3

 


When the numerator and denominator of a fraction are
switched, the resulting fraction is called the reciprocal.
All About Alice
• Find the missing value that will make each equation true.
( x 8 )( x n )  x 3
4 2
(a )
n
a
 y 
 10 
y 


n
1
 3
a
3
y
6
All About Alice
• In A Wonderland Lost, we found that the area of the
forest, which is decreasing by 10% every year, could be
calculated using the exponential function
f ( y )  1,200,000  (0.90)
y
The base value is 0.90 because the new rainforest area is
100% – 10% = 90% of the previous area.