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1.3.1 laws of exponents
供稿人: Salafia
(原作者: Warren)
An exponent is a small number written above
and to the right of another number. It represents
the number of times you multiply by the other number.
The larger number below the exponent that is used
to multiply is called the base.

The exponent and base together are called the power.
Exponents
23 = (2)(2)(2) = 8
base
23 = (2)(2)(2) = 8
exponent
8 is a power of two. It is the
third power of two.
23 = 8
power
10
Let's review:
104 = (10)(10)(10)(10) = 10,000
Which of these numbers is the base?

Pick One:
4
10,000
10
Let's review:
104 = (10)(10)(10)(10) = 10,000
Which of these numbers is the exponent?

Pick One:
4
10,000
10
Let's review:
104 = (10)(10)(10)(10) = 10,000
Which of these numbers is the power?

Pick One:
4
10,000
10
25
104 = 10,000

10 is the base.
4 is the exponent.
104 or 10,000 is the power.
Simplify:
25
16
32
2
25=(2)(2)(2)(2)(2) = 32
Now that we've had that review, let's look at
what happens when we multiply and divide
powers with the same base or the same exponents.

We'll develop some shortcuts to use so we can write
complicated expressions more simply, 
especially
when we're working with exponents.


Let's look at multiplication first.
#1: Multiplying Powers
with the Same Base
Simplify:(45)(46)
= ((4)(4)(4)(4)(4)) ((4)(4)(4)(4)(4)(4))
= (4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)
= 411
Let's look at a couple more examples before we
make a rule (pay close attention to the relationship
between the exponents in the beginning and the
single exponent at the end):

(32)(313)= ((3)(3)) ((3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3))= (3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)
= 315

(53)(54)
= ((5)(5)(5)) ((5)(5)(5)(5))= (5)(5)(5)(5)(5)(5)(5)
= 57
Keep the base and add the exponents
Keep the base and subtract the exponents
Multiply the bases and add the exponents
Add the bases and keep the exponents
Look at this example and pick a word-rule for
a shortcut to simplify these kinds of expressions.
(25)(23) = 28
(ab)(ac) = (a + a)bc
(ab)(ac) = (a)bc
Keep the base and
add the exponents!
What is the algebraic equation that
represents this rule?
The rule is:
(ab)(ac) = ab+c
(ab)(ac) = ab-c
When multiplying
powers with the
same base:
Keep the base and

add the exponents!
(ab)(ac) = ab+c
Simplify:(45)÷(42)
It's easier to see how this works when we
write the division as a fraction:
= 45    =   (4)(4)(4)(4)(4)   We can cancel two fours
   42                    (4)(4)         in the numerator and
                                        denominator.

(4)(4)(4)(4)(4)     =    (4)(4)(4)    = 43       
         (4)(4)
#2: Dividing Powers
with the Same Base
Let's look at a couple more examples before we
make a rule (pay close attention to the relationship
between the exponents in the beginning and the
single exponent at the end):

(37)÷(33) = (3)(3)(3)(3)(3)(3)(3) = (3)(3)(3)(3) = 34
                           (3)(3)(3)


(511)
÷(54)
= (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)

                (5)(5)(5)(5)

= (5)(5)(5)(5)(5)(5)(5)  =  57
Keep the base and add the exponents
Keep the base and subtract the exponents
Divide the bases and subtract the exponents
Subtract the bases and keep the exponents
Look at this example and pick a word-rule for
a shortcut to simplify these kinds of expressions.
(25)÷(23) = 22
Keep the base and
subtract the exponents!
(ab)÷(ac) = (a - a)bc
(ab)÷(ac) = (a)b÷c
What is the algebraic equation that
represents this rule?
The rule is:
(ab)÷(ac) = ab+c
(ab)÷(ac) = ab-c
When dividing
powers with the
same base:
Keep the base and
subtract the exponents!
(ab)÷(ac) = ab-c
#3: Multiplying Powers
with the Same Exponent
Simplify:(54)(64)
= ((5)(5)(5)(5)) ((6)(6)(6)(6))
=(5)(5)(5)(5)(6)(6)(6)(6)
=(5)(6)(5)(6)(5)(6)(5)(6)
=((5)(6))4
=304
Let's look at another example before we
make a rule (pay close attention to the relationship
between the bases in the beginning and the
single base at the end):

Simplify:(23)(53)

 

= ((2)(2)(2)) ((5)(5)(5))

 

=(2)(2)(2)(5)(5)(5)

 

=(2)(5)(2)(5)(2)(5)

 

=((2)(5))3

 

=103

Keep the base and add the exponents
Keep the base and subtract the exponents
Divide the bases and keep the exponent
Multiply the bases and keep the exponent
Look at this example and pick a word-rule for
a shortcut to simplify these kinds of expressions.
(39)(49) = 129
Multiply the bases and 
keep the exponent!
(ac)(bc) = (a + b)2c
(ac)(bc) = (a)bc
What is the algebraic equation that
represents this rule?
The rule is:
(ac)(bc) = (ab)c
(ac)(bc) = ab+c
When multiplying
powers with the
same exponent:
Multiply the bases
and keep the exponent!
(ac)(bc) = (ab)c
#4: Dividing Powers
with the Same Exponent
Simplify:(65)÷(25)
It's easier to see how this works when we
write the division as a fraction:

= 65    =   (6)(6)(6)(6)(6)   =   (6)  (6)  (6)  (6)  (6)
   25           (2)(2)(2)(2)(2)         (2)  (2)  (2)  (2)  (2)         

= (6÷2)5 = 35
Let's look at another example before we
make a rule (pay close attention to the relationship
between the bases in the beginning and the
single base at the end):

Simplify:(203)÷(53)

 

= 203    =   (20)(20)(20)   =   (20)  (20)  (20)

    53              (5)(5)(5)             (5)    (5)    (5)         

 

= (20÷5)3 = 43

Keep the base and add the exponents
Keep the base and subtract the exponents
Divide the bases and keep the exponent
Multiply the bases and keep the exponent
Look at this example and pick a word-rule for
a shortcut to simplify these kinds of expressions.
(509)÷(29) = 259
Divide the bases and
keep the exponent!
(ac)(bc) = (a - b)2c
(ac)÷(bc) = (a÷b)c
What is the algebraic equation that
represents this rule?
The rule is:
(ac)(bc) = (ab)c
(ac)(bc) = ab+c
When dividing
powers with the
same exponent:
Divide the bases and
keep the exponent!
(ac)(bc) = (a÷b)c
Simplify:(65)2
= (65)(65)
= ((6)(6)(6)(6)(6)((6)(6)(6)(6)(6))
= (6)(6)(6)(6)(6)(6)(6)(6)(6)(6)
=610
#5: Power of a Power
Let's look at a couple more examples before we
make a rule (pay close attention to the relationship
between the exponents in the beginning and the

single exponent at the end):

Simplify: (32)7= (32)(32)(32)(32)(32)(32)(32)
= ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3))
= (3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3) = 314

Simplify: (53)4
= (53)(53)(53)(53)
= ((5)(5)(5)) ((5)(5)(5)) ((5)(5)(5)) ((5)(5)(5))

= (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)
= 512
Keep the base and multiply the exponents
Keep the base and add the exponents
Keep the base and subtract the exponents
Multiply the bases and keep the exponent
Look at this example and pick a word-rule for
a shortcut to simplify these kinds of expressions.
(95)6 = 930
Keep the bases and multiply the exponents!
(ac)(bc) = (ab)c
(ac)÷(bc) = (a÷b)c
What is the algebraic equation that
represents this rule?
The rule is:
(ab)(ac) = (a)b+c
(ab)c = abc
Now you're ready to take the next quiz:
7.N.4 Part 2 - Applying the Laws of Exponents
See if you remember what to do.
Remember you can always retake the quizzes.
Don't forget to email me if you have a question:
awarren@portchesterschools.org
GOOD JOB!
THE END!
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