Download Operations with Algebraic Expressions

Operations with Algebraic
Expressions
Chapter 5
Vocabulary



Two terms which contain the same variable
or variables with corresponding variables
having the same exponents are called like
terms.
Two terms which contain different variables
or have different corresponding exponents
are called unlike terms.
Any operation which is completed to an
algebraic expression is known as simplifying.
Adding and Subtracting Algebraic
Expressions

The addition or subtraction of any algebraic
expressions is done as follows:




Identify the like terms.
Combine the like terms by adding or subtracting the
coefficients and do not change the powers.
Bring down any unlike terms.
Example: Find the sum of the following expressions.
 15abcand 6abc
 9abc
Practice: Simplify each expression

1.  12b2   5b2
 12b  5b
2
 7b

2. 3ab  5b  ab  4ab  2b
2
2
3. 6a  [5a  6  3a ]
6a  [2a  6]
8a  6
6ab  3b


4. 4 x2  2 x  3  2 x2  5x  3
4 x2  2 x  3  2 x2  5x  3
2x2  7 x
Product of Powers


When two or more powers with the same
base are multiplied it is known as the product
of powers.
a
b
a b
Product of Powers Rule:
x x x


The powers are added and the base is kept.
Example: Find the product of
y y y
3
5
y
8
35
y
3
and
5
y.
Power of Products


When a power is raised to a power it is
known as the Power of Products
a b
ab
Product of Powers Rule:
x x
 


The powers are multiplied and the base is kept.
Example: Find the result of
m  15m
5 3
m
53
m  .
5 3
Practice: Simplify each expression
1.
x y x y 
3
4
2
7
2.
4b c3bc 
7
6
5 11
x y
3.
ab 
2 4
8 7
12b c
4.
4 8
ab
 
5.  3x 5x
5
 15x
7
12
3v 
5 3
27v15
6.
 23g

9 2
 29 g
18
 18g

18
Quotient of Powers

When a power is divided by a power it is
known as the quotient of powers.
a

x
a b
Quotient of Powers Rule:
x
b
x


The powers are subtracted and the base is kept.
12
x
Example: find the result of
.
5
x
12
x
125
x
5
x
x
7
Negative Powers


When a power is negative it must be rewritten
using a positive power.
Negative Powers Rule:


x
a
1
 a
x
The negative power and base are moved to the
opposite part of the fraction and changed to a
positive.
Example: Simplify the expression x 3
1
x  3
x
3
Practice: Simplify each expression
1.
3.
m11
m5
21
w
m
1
21
w
6
5.
36 h
9
3h
6
12 h
3
12
h3
2.
25d 19
5d
5d 18
4.
2
v 4
2v 4
6.
9k 20
26
63k
1 6
k
7
1
6
7k
Multiplying Binomials


When two binomials are multiplied it is known
as a double distribute or a F.O.I.L.
Example: x  5 x  3


x x  3 
5x  3 

x  3x
2
5x  15
x  2 x  15
2
Practice: Simplify each expression
x  1x  4
1.
2.
x  3x  4
2
3. w  63w  9
2m2  11m  21
4.


h
3
 4h  8h  32
2
11  x 4  x 
44  15 x  x 2
3w2  27 w  54
5. h2  8 h  4
2m  3m  7
6.
3q  45q  4
15q  8q  16
2
Dividing a Polynomial by a Monomial.


When dividing a polynomial by a monomial, divide
each term of the polynomial by the monomial.
The resulting polynomial will have the same
number of terms as the original.
21a b  3ab
Example:
3ab
2


Example:
8a5  6a 4
2
2a
2
21a b 3ab

3ab 3ab
7a  1
8a5 6a 4
 2
2
2a 2a
4a  3a
3
2
Practice: Simplify each expression
1.
 36 y10  6 y 2
6 y2
 6y 1
8
3.
cm  cn
c
mn
5.
y2  5 y
y
 y5
24 x3 y 4  18 x 2 y 2  6 xy
2.
 6 xy
 12 x 2 y 3  3xy  1
8c 2  12 d 2
4.
4
2
2
 2c  3d
2
1
.
6
cd

4
.
0
c
d
6.
0.8cd
2  5c
Scientific Notation



Scientific Notation represents extremely large or
small numbers using powers of base ten.
The coefficient is the number which multiplies the
power and must be between one and ten.
Example: Express 25,000,000 in scientific notation.
2.5  10

7
or
2.5E 7
Example: Express .00000357 in scientific notation.
3.57 10
6
or
3.5E  6
Practice: Express in Scientific Notation
1.
3894000
2.
3.894 E 6
 5.81E 3
3. .00000079124
7.9124 E 7
5.
90236000
9.0236 10
 .00581
7
4.
7584000000000
7.584 1012
6.  00000000002359
2.359 E 11