Download 2-6-3 Exponent Rules

2-7-1 Number Properties
Properties are the rules of the game. If the rules are changed a different kind of math is played.
These properties are for everyday math.
Commutative
property of addition.
Commutative property of
multiplication.
a+b=b+a
axb=bxa
7+8=8+7
(-3)+4=4+(-3)
7x8=8x7
(-3)4=4(-3)
The commutative property says the order
doesn’t matter for addition.
The commutative property says the order
doesn’t matter for multiplication.
Associative property
of addition.
(a+b)+c = a+(b+c)
(7+8)+2=7+(8+2)
15+2 = 7 + 10
Associative property
of multiplication.
(axb)xc=ax(bxc)
(7x8)x2=8x(7x2)
56x2=8x14
The associative property says the
grouping doesn’t matter for
addition.
The associative property says the
grouping doesn’t matter for
multiplication.
An identity for a particular operation doesn’t change the identity of the number when the operation is
done.
7 + 0 =7
Additive Identity
a+0=a
Zero plus a number is that number.
8x1=8
Multiplicative Identity 1xa=a
One times a number is that number.
An inverse for a particular operation is the number that returns the identity.
Additive inverse
a + (-a) =0
Multiplicative inverse
1
a   1
a
The additive inverse is the negative of
the number.
7 + (-7) =0
-7 is the
additive inverse
of 7
3 x 1/3 = 1
The multiplicative inverse of a rational
number is it’s reciprocal.
3x(4x)=3(4)xx=12x2 using both the commutative and associative properties of multiplication.
Distributive
property
a(b+c)=ab+ac
53  4  5  3  5  4
Multiplying a sum by some number
is the same as multiplying each
term by that same number.
3x2 x  8  6 x 2  24 x
Practice: List the property illustrated by the following example.
a) 5+0=5
R+6=6+R
3+(7+8)=(3+7)+8
b) 4t(7t8t)= (4t7t)8t
c) xy=yx
4
4
4 9  2  4  (9)  4  2
0+t=t
2z+7=7+2z
3x=3x+0
1x=x
7(2x-8)=14x-56
d)  m  m
2 3
4
5
1
2 1   1 1
3 4
5
6
5
e) -8d+8d=0
0-7y=-7y
40
8+(-8)=0
9t  12 
3
12t  16
4
15y+10=5(3y+2)
1
4  1
4
-2e(3e-7)=-6e2+14e
2-7-2 Distributive Property
To illustrate the distributive property look at the problem 3(2x-7).
This problem says 3 times 2x-7 or 3 of (2x-7). Multiplying by 3 means add 2x-7 over and over.
2x-7+2x-7+2x-7 Simplify this by using the commutative property. 2x+2x+2x-7-7-7=3(2x)+3(-7)
To use the distributive property, multiply each term inside the parenthesis by the multiplier. Study the
following examples. Look where exponent rules are used. Watch the multiplication when negatives are
involved. After the distribution, combine any like terms.
Examples:
5  3x  8   15x  40
3x  7  6 x   21x  18 x 2
  5s  8   5s  8
4 x  2  3x  9   4 x  6 x  18  2 x  18
3w2 y3  7w  5 y 2   21w3 y3  15w2 y5
5  3  6x   5  3  6x  2  6x
There can be more than two terms inside the parenthesis.
3 4  6d  8d 2   12  18d  24d 2
4 z 2  6  5 z   24 z 2  20 z 3
5 7 x2  5x  9  35x2  25x  45
6 xy 2  5x2 y  2 xy  3xy 2   30 x3 y3  12x2 y3  18x2 y 4
3gh3   7 gh  5gh3  6 gh2   3gh3  7 gh  5gh3  6gh2  8gh3  7 gh  6gh2
16 x 2  12 x 16 x 2 12 x


 4 x 2  3x
4
4
4
16 x 2  12 x 16 x 2 12 x


 4 x 2  3x
4
4
4
15w3  25w5  75w4 15w3 25w5 75w4
5w 2



1
 5w
15w3
15w3 15w3 15w3
3
Practice:
a)
3 2 x  5  
25  2 y  3 
7  x  8 
b)
53  8x  
3 3  7 x  
100  2  5 x  
c)
  4  2x  
  5x  8 
4  2 y  
d)
4  3x  8  
10  5 x  15  
2  7  8y  
e) 5 y  7  6 y  

f) 3 j 6 j  5 j
3

8h  7h  6  
2
g)  9  5a  8a

4m2  m  5  

  4l  12  
2
2
3
k 12  5k  
 1 2 z 2  6  10 z  
 u  8 
41
h)
3x   3x  9  
9h  2  5h  8  
i)
2 y5 z 3  7 y  2z 2  
j)
8   5  9x  
5 2 3   3 7 8  9x  
k)
8  5  3x  9  
9  5  x  5  
13  7  3x  9h  
9  3w  6  8w2  
  q 2  6q  8 

l) 8 2  6 x  5 x

m) 4 10  4y  y

2
2
3


3
n) 2x y x y  xy  xy
2

3
o) uv u v  uv  uv
2
2
2
4
2

mn3 8m5  5n2  
5
15x
2
4
3
16 x 2  12 x
s3t 2  2 3 s 2t 2  6 5 st 2  3st 2  
5

3
3

2
s)
2
2

a 2  2a

 25

10 x 2  10 x

100
6

r) 4a b  10 gh  5g h  gh
2
x 2  12 x

1
2
6e2  12e
2
30v 3  21v 5  18v8

3v 5
2
2

31
5 7u 2  8u  1 
25 p 3  15 p

5

q) gh  gh  gh  gh
 vw8

15v w 
 v2  
 5

10   6 x  9  
12
 5x  9 
1
p)
9x   7 x  5 
7
12
3
xy3   3xy  3 4 xy3  xy 2  
v2 w3   vw  38 v2 w3  38 vw2  
15h3  25h51  75h 25

5h31
35k 5  21k 4  77k 7
 7k 3
15u 3  5u 5  75u 4

3u
3i 5 j 3  21i 4 j 5  i 7 j 3

u)
3i 2 j 3
3x5 y 3  21x 4 y 5  12 x 7 y 3

9 x5 y 2
5w3v 3  w4v 4  w7v 3

v)
wv 5
3e4 f 13  21e2 f 5  e5 f 5

e 2 f 5
t)
42
2-7-3 Factoring
Assume 12x+6 is the result of distribution. What did it look like before the distribution was done?
There was something outside a set of parenthesis and two terms inside the parenthesis.
(
+
)
6
(
2x
+ 1
)
Separating an expression so that the smallest possible pieces called factors multiply together to get the
original expression is called factoring.
Look at each term and find the largest factor that is in all terms.
Example: 72 x  60 x  36 Find the largest number that divides evenly into 72, 60 and 36.
This number goes in front of the parenthesis.
72 x 2 60 x 36
2
Divide each term by the common factor.
Fill in the positions in the parenthesis. Distribute
to check the factoring.
24x2 y 2  16x2 y  56xy 2
12
,
,
12
12
(
12
(
12
6x2
+
+
)
+ 5x
+ 3
)
The largest factor in all terms is 8xy. When the division is done there
shouldn’t be any negative exponents. Notice the negative in the third term.
Divide each term by the common factor.
8xy (
24 x y
16 x y 56 xy
,
,
8 xy
8 xy
8 xy
2
2
2
+
-
)
+ 2x
-
7y )
2
8xy (
3xy
The division can be done in your head, but some students need to write it down.
Practice: Factor the following.
a) 15y-25=5(3y-5)
14z+56=
39u-13=
81p-18=
7t+21=
b) 8+8t=
14-21w=
9v-81=
2x-10=
3w+9=
c) -2/3y-10/3=-2/3(y+5)
Note the negative.
4y 8
 =
5 5
10 15r

=
12 12
d) 6x2-9x+12= 3(2x2-3x+4)
14e2+21e+56=
2x2-4x-8=
8a2+4a+4=
10b2+5b-25
-5-5t-5t2=
-8+16e-24e2=
12-24y2+36y=
15r+15r2-25
bx-7x=
3s-3ts=
4ef+3f=
2w-wx=
14y2+21y=
81z+18z2=
10q2-20q=
8a-8a2=
12xy2+16xy+24x=
a2x2+a2x+a2=
2a2x-4a=
e3-5e2x=
e) 3+3w+3w2=
f) ax+2a=a(x+2)
g) 4x2-12x=4x(x-3)
h) 2ax2+4ax-8a=
i) 3x3y2+9x2y2-12x2y3=3x2y2(x+3-4y)
4
25
y6
25

5r 2
 
3 3
25a4b2+45a2b3-15a2b5=
43
j)
a7b5  5a8b7  a5b4 
k) 25a4b2 + 75a2b3 - 100a2b2
l)
3
44
x 1
 4
4x 
4
2
Hint: Factor out a ¼.
2e4 f 3  4e2 f 2  ef 3 
21x3y6+14x2y5-28x4y4
15a 2 21a

8
8