8.1 Zero and Negative Exponents:
Use a Graphing Calculator to fill in the following table:
A
21 = _____
B
2-1 = ____
22 = _____
2-2 = ____
23 = _____
2-3 = ____
C
1
= ____
21
1
= ____
22
1
= ____
23
D
1
= ____
2−1
1
= ____
2−2
1
= ____
2−3
E
20 = ____
1
= ____
20
(-3.2)0 = ____
Which pairs of columns are the same?_______________________
Negative Exponent Rule:
Ex:
6-4 = ____
Ex:
60 = ____
For every nonzero number a and integer n,
𝑎−𝑛 =
1
𝑎𝑛
Zero as Exponent
For every nonzero number a,
𝑎0 = 1
Ex 1: Simplify WITHOUT USING A CALCULATOR!
4-3
Ex 2: Simplify WITHOUT USING A CALCULATOR!
a. 4yx-3
b.
1
𝑤 −4
8.3 Multiplication Powers with the Same Base
Fill in the blanks below:
Factors
21·21
22·22
Product Using Repeated Factors
2·2
(2·2) · (2·2)
22
24
Power of 2
22·23
(___·____)· (____·____·____)
2__
24·22
(___________)·(______)
2__
Multiplying Powers with the Same Base
Ex1 32 · 34 = 3----
For every nonzero number a and integers m and n,
Ex2 x3 · x2 · y2 = x3+2·y2
am · an = am___n
Ex 3 Simplify
=x__y2
5x·2y4·3x8
Factors
(22)3
22·23
Product Using Repeated Factors
(2·2) ·(2·2) ·(2·2)
(2·2) · (2·2·2)
2__
2__
Power of 2
(23)3
(___·___·___)· (___·___·___)· (____·____·____)
2__
23·23
(___________)·(_________)
2__
Raising a Power to a Power
Ex1 (54)2 = 54__2 = 58
For every nonzero number a and integers m and n,
Ex2 (x2)5 = x__
(am)n = am__n
Ex 3 Simplify
c5(c3)-2
Fill in the table and simplify:
Factors
23
22
24
22
25
22
Product Using Repeated Factors
2×2×2
2×2
2×2×2×2
2×2
___ × __ × __ × __ × __
__ × __
26
22
(__________________________)
____________________
Power of 2
21
2__
2__
2__
Dividing Powers with the Same Base
For every nonzero number a and integers m and n,
𝑎𝑚
𝑎𝑛
Ex1
37
32
= 37__2 = 3___
= 𝑎𝑚___𝑛
Ex2 Simplify
𝑐 −1 𝑑3
=
𝑐 5 𝑑4
=c___-____d____-____
𝑑 __
= c__d___ = 𝑐 __
Fill in the table:
Factors
2 2
( )
3
2 3
( )
3
Product Using Repeated Factors
2
2
2__
( ) × ( ) = __
3
3
3
2
2
2
2__
( ) × ( ) × ( ) = __
3
3
3
3
2
22
22
22
2__
(3)
( )×( )= 2
3
3
3
Raising a Quotient to a Power
4
9
𝟒 𝟑
𝟓
Ex1( ) =
For every nonzero number a, and b and integer n,
𝑎 𝑛
Final
𝑎𝑛
(𝑏 ) = 𝑏𝑛
𝟒 __
𝟓__
=
𝟒 𝟑
Ex2 Simplify (𝒙𝟐)
𝑎 −𝑛 𝑏 𝑛
=( )
𝑏
𝑎
Lemma … ( )
Ex3 Simplify (−
𝟐𝒙 −𝟒
)
𝒚
NOT IN BOOK
1
3
Non Integer exponents:
Ex1 273 = √271 = 3
For every nonzero number a, and integer m, and n,
Ex2 𝑥 3 =
2
𝑚
𝑛
𝑎 𝑛 = √𝑎𝑚
You’ve got a Gambling Problem:
Your betting strategy is “double or nothing” (known sometimes as the Martingale Betting Strategy).
Using this betting strategy means that you double your money each time you win. When you lose, you
lose all your money. So, you either double the amount your betting or lose it all. Hence the name
“double or nothing.”
a. If you start with $1 how much will you have after winning once? Twice in a row? Three times in a
row?
b. How many times in a row will you have one if you have $27?
c. What if you start with 5 dollars? How much will you have after winning once? Twice in a row? Three
times in a row?
d. How many times in a row will you have one if you have $5·21? $5·22? $5·210?
e. Come up with a general formula for the “double or nothing” betting strategy where A is the original
amount you start with, n is the number of times you win in a row, and T is the amount you win.
Exponential Functions (y = a·bx)
An exponential function is a function of the form _________, where a is a nonzero constant b is
greater than 0 and not equal to 1, and x is a real number.
Ex1 Fill in the table, then graph the functions on the same graph below, each in a different color.
a. y = (1.5)x
Domain (x)
-2
0
1
2
3
b. y = (.5)x
y = (1.5)x
1.5-2
Y
.44
c. y = -3 (1.5)x
Domain (x)
-2
0
1
2
3
y = 3(1.5)x
Y
Domain (x)
-2
0
1
2
3
y = (.5)x
y
What happens to the y value of
exponential function as x increases if
the base is between 1 and zero?
_______________________________
_______________________________
_______________________________
Growth v. Decay
Exponential Growth can be modeled with the function y = a·bx for a > 0, and b > 1.
Starting amount when x is _____.
y = a · bx
A base which is greater than one is a _____ ___ ______.
Exponential Decay can be modeled with the function y = a·bx for a > 0, and 0<b<1.
Starting amount when x is _____.
y = a · bx
A base which is greater than one is a _____ ___ ______.
Ex 1 After a milk container has gone bad bacteria grows by 10% every week. If you start with 20 million
harmful bacteria at the point the milk goes bad, how many harmful bacteria will their be in a week? Is
this growth or decay? Is b > 1 or 0<b<1?
Ex2 You have had 2000 dollars set aside into an account when you turned 3 years old for a college fund.
The fund gains 10% interest compounded biannually (twice a year). This means that you collect interest
on it twice a year at half the rate of 10%...5% (see page 477). How much money will you have when you
turn 18? Is this growth or decay? Is b > 1 or 0<b<1?
Ex3 The half-life of a radioactive substance is the length of time it takes for one half of the substance to
decay into another substance. To treat some forms of cancer doctors use radioactive iodine. The halflife of Iodine-131 is 8 days. A patient receives 12 mCi (millicuries, a measure of radiation) treatment.
How much iodine 131 is left in the patient after 16 days? Is this growth or decay? Is b > 1 or 0<b<1?