Transcript Document

1. Be able to divide polynomials
2. Be able to simplify expressions involving powers of
monomials by applying the division properties of powers.
3
4x y
2 xy
3
 3 xy

3
 yx
2



5
8
2
2
36a b ab

ab
6
Monomial: A number, a variable, or the product of a number
and one or more variables
Constant: A monomial that is a real number.
Power: An expression in the form xn.
Base: In an expression of the form xn, the base is x.
Exponent: In an expression of the form xn, the exponent is n.
Quotient: The number resulting by the division of one number by another.
Repeated multiplication can be represented using exponents.
3  x  x  x  x  y  y  z  3x y z
4
2
To expand a power, use the exponent to determine the number of
times a base is multiplied by itself.
82 x 3  8  8  x  x  x
Product of Powers: When two numbers with the same base are
multiplied together, add the exponents and leave the base unchanged.
a a  a
m
n
m n
Power of a Product: In a product raised to a power, the exponent
applies to each factor of the product.
3xy
2
3 x y
2
2
2
Power of a Power: When a power is raised to another power,
multiply the exponents and leave the base unchanged.
a 
m n
a
m n
Remember: Follow the order of operations when applying more than
one property!
3x
2
 
3
 x y  3x
3
23

3
 
y 3 x
3
5 3
15 3
53 3

27
x
y

27
x
y
y
3
3x 3
Simplify: 2
x
Step 1: Rewrite the expression in expanded form
3x 3 3  x  x  x

2
x
x x
Step 2: Simplify.
3 x  x  x
 3x32  3 x
x x
Remember: A number divided by itself is 1.
For all real numbers a, and integers
m and n:
am
mn

a
an
 3x 
Simplify: 

 y
2
Step 1: Write the exponent in expanded form.
2
 3x 
3x 3x
  

y y
 y
For all real numbers a and b, and
integer m:
Step 2: Multiply and simplify.
m
9x
3 x 3 x 3x  3x 3 x
 
 2  2
y
y
y y
y y
2
2
2
a
a
   m
b
b
m
6
3
33
6 3
3
3
3
Apply quotient of
powers.
 4
 
 x
2
 3x 
 2 
 x y
4
3
4
x2
 3 x 42 


 y 
16
x2
33 x 2
y3
2
Apply power of
a quotient.
 
3
Apply quotient of powers
3
33 x 23
y3
27 x 6
y3
Apply power of a quotient
Apply power of a power
Simplify
1.
 2a b 


 a 
2.
 4x y

 2 xy
2
3
3
3
  2
   
  xy 
2
 2a b 


 a 
2
3
3
 2a b 
2 1 3

  2a b
 a 
2


 2ab
3
 23 a 3b3
 8a3b3
 4x y

 2 xy
3
 4x y

 2 xy
3
3
3
  2
   
  xy 
2
2
  2   4 31 31  2 2
       x  y   2 2
 x y
  xy   2
4
 2x2 y 2  2 2
x y


2x2 y 2  4

x2 y2
 4 2 x
8
2 2
THINK!
y
2 2
x2-2 = x0 = 1