5.1 Exponents and Their Properties
Recall, when given the expression ab
a is referred to as the base
b is referred to as the exponent or power
It is the value of the power that dictates how many times the base is to be multiplied by
itself.
ab means a times itself b times
example: 43 means _______________________
While we’re at it, let’s go over some problems that can cause trouble.
21
20
22
23
2/2
-21
-20
-22
-23
2/1
(-2)1
(-2)0
(-2)2
(-2)3
1/2
(-21)
(-20)
(-22)
(-23)
0/2
2/0
-2/0
-2/2
-2/1
-2/-2
We’ll talk about this later.
Let’s make a quick table (handy reference):
expression
20
21
22
23
24
meaning
value
expression
25
26
27
28
29
meaning
value
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Discovering the Product Rule:
23  2 4
Starting with
it means
 2  2  2   2  2  2  2
which equals
8  16
which equals
128
which is the same as
Notice that the final answer is given in
“exponential notation.”
27
What do you notice? How does the original problem (given in exponential notation)
compare to the answer?
Discovering the Quotient Rule
This -->
means this -->
which equals this -->
which equals this -->
25
23
22222
222
Notice that the final answer is given in
32
“exponential notation.”
8
4
which is the same as this --> 22
What do you notice? How does the original problem (given in exponential notation)
compare to the answer?
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Let’s assume that you now know and love the Quotient Rule. We’re going to use this
concept to address the 20 issue. We’ll do a few problems to lead up to and conclude
what the answer is to a number raised to a zero power.
25
We know 2  25 2  23
2
What if the numerator and denominator are the same?
25
 255  20 Well, that's nice but where does the 20 =1 come in?
5
2
Well don't we know that any number divided by itself equal 1?
Ta Daaaaa!
Just like magic!
Then we can safely conclude that:
25
 1 and 255  1 and, finally, 20  1
5
2
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Discovering the Powers to Powers Rule
Powers to Powers – Sounds Powerful, Huh? Well, it is! First, back to basics. We know
that an exponent tell us how many times to multiply the base by itself.
x 3 means x  x  x
Well, what if x  y 2 ? Just substitute y 2 for x to get
 y2 
3
y 
2 3
Note that the ( ) are critical here.
means y 2  y 2  y 2 which we know equals y 6 (based on the product rule).
Therefore,  y 2  must equal y 6 !
3
 y 2   y 6 What do you notice about that?
3
Discovering the Raising a Product to a Power Rule
If we had  4a 2b  , it means  4a 2b   (4a 2b)  (4a 2b).
3
All of the above is multiplication, which we know is communitative
(meaning the order in which they are multiplied doesn't matter. Soooo.....
 4a b   (4a b)  (4a b)  4  a
2
2
2
2
• b  4  a2 • b  4  a2 • b
 4 3 • a 6 • b3
 4 3 a 6b 3

 4a b 
2
3
 43 a 6b 3
A product to a power....what do you think?
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Moving along the same lines, we have the following:
Homework:
HIH #1 5.1: 44, 72, 86
MML/Quiz due: ____________ Text HIH due: ____________
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