Transcript 1 - Mr. Hood

6.3 Division Properties of
Exponents
When working with division problems involving exponents,
you can expand and cancel.
1 1
1
x /
x/
xxx
x5 /

3
/x  /x  x/
x
 x2
You can cancel because
any number or variable
divided by itself is
equal to one.
Can you expand and cancel this in your head?
When you are expanding and canceling the exponents in
your head, what mathematical operation are you using?
Subtraction
When working with division problems involving exponents,
you can expand and cancel.
1 1 111
3 4
3
xxxyyyy
// //
/

///
9x 7 y 3
/  3xxxxxxxy
/
3x y

y3
3x 4
Once you have practiced
this, you may expand and
cancel in your head!
Here’s how…
Expand and cancel in your head!
3x3 y 4
1 y3

4
7
9x y 3 x

y3
3x 4
What is
left and
where is it
located?
Example 1 Simplify.
1. Write problem.
39
4

3
35
 81
2. Expand and cancel (you
may do this in your head).
What is left and where
is it located (numerator
or denominator)?
You cannot
3. Simplify.
cancel the
base!
Example 1 Simplify.
You cannot cancel the
bases to one when the
powers are different!
19
9
3
1
5  5
3
1
 14
1
Wrong answer!
Example 2 Simplify.
x4
1
 3
7
x
x
What is left and where
is it located (numerator
or denominator)?
You cannot
have a
denominator
without a
numerator!
Example 3 Simplify.
12 m5 4m 4

1
3m
 4m 4
What is left and where
is it located (numerator
or denominator)?
A nonzero number to the zero power is 1.
258
8 8

25
258
 250
1
x5
55

x
x5
 x0
1
Zero to the zero power is undefined!
Example 4 Simplify.
25x3

3 5 1
5x
5
5
1
3
25x
5
3
5x
Example 5 Simplify.
53 y 6
5y
6

521
 25
What is left and where
is it located (numerator
or denominator)?
Example 6 Simplify.
 83 x 4 y3 1 x2

3 2 7
  8 x y
y4
x2
 4
y
What is left and where
is it located (numerator
or denominator)?
Power of a Quotient Property
To find a power of a quotient, find the power of the
numerator and the power of the denominator and then
2
divide.
4
4
2
2
2





2
2x


2
x

2

 
4 4
  



 
4
w
 w 
y2
 5  52
 y 
16
16
22 x2

 4
 2
25
w
y
4 x2
 2
You must use
You must use
y
parentheses!
parentheses!
Example 7 Simplify.
2
 42 
4

     2 
x 
x
 
The
exponent
applies
only to the
base!
16
 2
x
Example 8 Simplify.
2 3
 
3
2
 3x
3x


 y   y3


33 x23

y3
Must use
parentheses!
27x6
 3
y
Simplify.
Change the position of
all negative exponents –
keep the positive powers
where they are given.
77
a 5b
b
5

a
3 44
a bb
b11
 8
a
Example 9 Simplify.
x 77yy66
x 22
2
x y
x


y8
x9
Change the position of all negative
exponents – keep the positive powers
where they are given.
Example 10 Simplify.
3
6
4x3 yy6z
Change the position of 4x

4
2

4
all negative exponents – 32xy z
keep the positive powers
1 4
1
z
where they are given.

8 x13y26
z5

8x 4 y8
Expressions Involving Fractions
Change the position of
22
3
x3
3x
x3
x
all negative exponents –
 3
55 
55 3  x


x y
2y
2x
keep the positive powers xyy
where they are given.
3 5 5
3x
y
Multiply across the top (numerators)
 1 2 3
2x y
and across the bottom (denominators).
Expand and cancel.
What is left
and where is
it located?

3x8 y5
2x3 y3
3 x5 y 2

2
3x5 y2

2
Example 11 Simplify.
3x3 y 12x2 y2 36x5 y3


3
4x
y
4x y3
 9 x4
What is left and
where is it
located?
When you are expanding
and canceling the exponents
in your head, what
mathematical operation are
you using?
Subtraction
Example 12 Simplify.
3
y2  y 
y2 y33

3  4   33  12
 x  xx xx 12
x

Change the position
of all negative
exponents – keep
the positive powers
where they are
given.

y2

x15 y2
y3
x15

y
8-A3 Pages 465-466 # 26–46 even, 49-57 all.
Algebra
rocks!
Which expressions are equivalent to
5-4 · 53
A.
5-8
5-8 · 53
B. -4
5
5-8 · 52
C.
5-3
5-8 · 54
D.
5-3
5-3 · 57
E.
53
5-4 · 52
F.
53
1
5
:
You can cancel to one when the value in the numerator and
the denominator are the same.
34
1
4
3
3
1
3
However, when the exponents are not the same you cannot
cancel the bases. The different exponents simplify to
different numbers.
34 81

3
27
3
3