CHAPTER 1-REVIEW
I) SECTION 1.4: OPERATIONS WITH
FRACTIONS
OBJECTIVES: You will learn
1)Multiply fractions and mixed numbers
2) Divide fractions and mixed numbers
3) Add and subtract fractions and mixed
numbers with the same denominator.
4) Find the least common multiple (LCM) of
two natural numbers
5) Add and subtract fractions and mixed
numbers with different denominators.
6) Applications
3
b) 3  10
8
3) Adding and subtracting fractions with the
same denominator
1) Multiply fractions:
Ex3: Add or Subtract
3 8
a) 
5 5
Ex1: Multiply. Your answer should be in
lowest terms
5 1
a)  3  2
9 6
b)
3

b)  5    6 
10 

2) Dividing fractions:
9 21

16 16
4) Find the least common multiple (LCM)
of two natural numbers
-Find the prime factorization of each
number
-Find the common factors of the numbers
-Multiply the common factors by the
remaining factors of the numbers
Ex4: Find the LCM of the given numbers
a) 9, 25
Ex2: Divide. Your answer should be in
lowest term
1
3
a)  7  13
9
8
1
b)8, 10, 16
5) Add and subtract fractions and mixed
numbers with different denominators
-Find the LCM of the denominators
-Rewrite each fraction as an equivalent
fraction whose denominator is the LCM of
the original denominators
-Add or subtract the numerators, keep the
common denominator
-Simplify to lowest terms, if possible
Ex5: Simplify. Your answer should be in
lowest terms
1
a)  6  5
5
d)
10 13 8


21 18 15
6) Applications:
Ex6: Word problems
a)Nancy gave birth to twins. One of the
7
babies weighted 5 lbs at birth, and the
8
1
other baby weighed 6 lbs . Find the total
4
weight of the twins at birth
5
1
b) 3  2
9
6
21
fluid ounces of a
40
3
solution. If she needs fluid ounces of the
8
solution for an experiment, how much of the
solution will remain?
b) A chemist has
c)
6  8
  
7  15 
2
c)Ross makes a batch of hot source that will
3
be poured into bottles that hold 6 fluid
4
ounces. If Ross has 108 fluid ounces of hot
source, how many bottles can he fill?
c)
2 2 2
 
5 5 5
Ex2: Simplify the given expression
3
a)  
2
3
b) 0.2 4
2
d) A pancake recipe calls for 1 cups of
5
whole-wheat flour to make 12 pancakes.
How much flour would be needed in order
to make 45 pancakes?
c)  3 2  (2) 3
 
d)  2
II) SECTION 1.7: EXPONENTS AND
ORDER OF OPERATIONS
OBJECTIVES: You will learn
1)Simplify exponents
2)Use the order of operations to simplify
arithmetic expressions.
1)Exponents:
a) Base, exponent.
Ex:
Ex1: Rewrite the given expression using
exponential notation
a) (5)( 5)( 5)( 5)( 5)

 
3 
3
3

4
2
2) Order of operations
Remove group symbols: (),{},[], perform in
the order they appear inside out.
Perform any operations involving exponents
Multiply and divide: perform in the order
they appear from left to right
Add and subtract: perform in the order they
appear from left to right
Ex3: Simplify the given expression
a) 17.1  8.58  3.9
b)  2  2  2  2  2  2
3
b) (2) 3  4(5)( 3)
g)
(4  5  20  10)( 5 2  2  15  10 2  2  51 )
c) 3  5(6 2  4  2  7)
III)SECTION 1.8: INTRODUCTION TO
ALGEBRA
8 4 1
  
d)
25  15 3
3 7

5  18
OBJECTIVES:
1)Build variable expressions
2)Evaluate algebraic expressions
3)Understand and use the commutative and
associative properties of real numbers
4)Use the distributive property
5)Simplify variable expressions
6)Identify terms and their coefficients
1)Build variable expression: a variable
expression is a combination of one or more
variables with numbers or arithmetic
operations.
Ex:
e)
3  5  7  4  23
1  3  17
Expression Different phrases
x+5
The sum of a number and 5
A number plus 5
A number is increased by 5
5 more than a number
The total of a number and 5
x-10
f)  4 2  17  (6) 2  5  10
10 less than a number
The difference of a number and 10
A number minus 10
A number decreased by 10
4
8x
The product of 8 and a number
8 times a number
8 multiplied by a number
x/6
The quotient of a number and 6
A number divided by 6
The ratio of a number and 6
Ex1: Build a variable expression for the
following phrases:
a)Twice a number
b) ( x  h) 2  3( x  h)  12 for x=-2, h=0.1
3) Properties of real numbers
a) Commutative property: for all real
numbers a, and b
b)One number divided by another
c)Six times the sum of two numbers
a+b=
ab 
Ex:
d) Half the difference of the number and 24
b) Associative property: for all real
numbers a, b and c
e)A professional baseball player is
appearing at a baseball card convention. The
promoter agreed to pay the player a flat fee
of $26,000 plus $21 per autograph signed. If
we let a represent the number of autographs
signed, build a variable expression for the
amount of money the player will be paid
f)A college charges $525 per credit for
tuition. If we let c represent the number of
credits that a student is taking, and if a
student has $1250 in expenses, build a
variable expression for the amount of money
students would pay for the semester.
2) Evaluate variable expressions
Ex2: Evaluate the following algebraic
expression under the given conditions
a) b 2  4ac for a= -2, b=-4, c=-3
( a  b)  c 
(ab)c 
Ex:
4) Distributive property
For all real numbers a, b and c
a (b  c) 
Ex:
5) Simplify variable expressions:
Like terms: are terms that have the
same variable factors with the same
exponents, or that are both constants
Ex:
Ex3: Simplify, where possible
5
a) 12(5  2 x)
CHAPTER 5: EXPONENTS AND
POLYNOMIALS
b) 12 x  23 y  124
SECTION 5.1: EXPONENTS
c)  3(3x  9)  3(12  4 x)  12
OBJECTIVES:
1)Use the product rule for exponents
2) Use the power rule for exponents
3)Use the power of a product rule for
exponents
4)Use the quotient rule for exponents
5)Use the zero exponent rule
6)Use the power of a quotient rule for
exponents
7) Evaluate functions by using the rules for
exponents
8) Applications
d)  2(6a  2b  5c)  3(3b  8c)
6) Identify terms and their coefficients
a) A term is either a number, a variable, or
a product of a number and variable
Ex:
b) A coefficient is the numerical factor of a
term
Ex:
Ex4: For the following expressions,
i)Determine the number of terms
ii)Write down each term; and
c)Write down the coefficient for each
Be sure to simplify each expression before
answering.
a) 8 x 3  3x 2  4 x  17
1)Product Rule: for any base x,
xm  xn 
Ex1: Simplify
a) m13  m 22
b) b 2  b 7  b 5
c) (4 x  2)12  (4 x  12) 9
d) (5 x 8 y 11 )(3x 5 y 13 )
Ex2: Find the missing factor
a) x 8  _____  x 17
b)_______  a 11  a 22
A word of caution:
x 3  x 7  _____, not____
b) 4(3x  5 y  5c)  (6 x  9 y  15)
2) Power rule: for any base x,
( x m ) n  _________
6
Ex3: Simplify:
a) (b12 ) 5
b) ( x 2 ) 8 ( x 5 ) 3
a)
x 13
x4
b) z 89  z 87
Ex4: Find the missing exponent
a) ( x 9 ) ____ ( x11 )12  x195
b) ( x
) p
___ 8
104
A word of caution:
(x 3 ) 7  ____, not_____
3) Power of a product rule: for any bases
x,y
( xy) n  _______
Ex5: Simplify
a) (10 x 2 y 5 z 2 ) 3
c)
(2 x  15)11
(2 x  15) 6
d)
35r 10 s 8 t 14
7r 5 s 6 t 9
A word of caution:
a 21  a 3  a 18 , not _____
Ex7: Find the missing exponent(s)(Assume
all variables are nonzero)
a) y ____  y 25  y 11
x 17 y ____
b) ____ 27  x 9 y 8
x
y
b) (a 11b17 c 20 ) 2 (a 10 b 8 c 3 ) 8
Ex6: Find the missing exponents
(2s ____ t ____ ) 6  64s 24 t 78
A word of caution: This property applies
only when we raise a product to a power,
not a sum or a difference.
( x  y) 2  ___________
5) Zero Exponent Rule: for any base x,
x 0  _____
Ex8: Simplify (Assume all variables are
nonzero)
 4
a)  
 15 
0
b) (22 x 5 y 12 z 9 ) 0
( x  y) 2  ___________
6) Power of a Quotient Rule: for any bases
4) Quotient Rule: for any base x,
xm

xn
n
 x
x and y,   
 y
Ex9: Simplify (Assume all variables are
nonzero)
Ex6: Simplify (Assume all variables are
nonzero)
7
 x8 y 5
a)  4 6
 2y z



3
 17 x 4 y 15 z 23 

b) 
11 34 32 
 32a b c 
0
b)The area of a circle with radius r is given
by the function A(r )  r 2 . Use the function
to find the area of the base of a circular
storage tank if the radius is 14 feet long. Use
  3.14
Ex10: Find the missing exponent(s). Assume
all variables are nonzero)
 a 3b 6
a)  8
 c



_____
 ab 9 
b)  5 11 
c d 
SECTION 5.2: NEGATIVE
EXPONENTS; SCIENTIFIC
NOTATION
______
7) Evaluate functions by using the rules
for exponents
Ex11: Evaluate the given function
a) f ( x)  x 5 , f (a 4 )
b) f ( x)  x 3 , f (2)
OBJECTIVES:
1)Understand negative exponents
2)Use the rules of exponents to simplify
expressions containing negative exponents
3)Convert numbers from standard notation
to scientific notation
4)Convert numbers from scientific notation
to standard notation
5)Perform arithmetic operations using
numbers in scientific notation
6)Use scientific notation to solve applied
problems
1)Negative Exponents: For any nonzero
base x, x n 
8)Applications:
Ex12:
a)The number of feet travelled by a freefalling object in t seconds is given by the
function f (t )  16t 2 . If a ball is dropped
from the top of a building, how far would it
fall in 4 seconds?
Ex1: Rewrite the expression without using
negative exponent (Assume all variables
represent nonzero real numbers)
a) 2x 10 
b)  12m 5 =
c)
x7
=
y 6
d)
 10 x 3 y 6
z 4 t 5
8
A word of caution:
2 3  ______, not _______
 x 8 y 5
h) 
6
 wz



8
2) Using the Rules of Exponents with
Negative Exponents
Ex2: Simplify the expression. Write the
result without using negative exponents
(Assume all variables represent nonzero real
numbers)
a) x 7  x 15
3)Convert numbers from standard
notation to scientific notation
To convert a number to scientific notation,
we write it in the form a  10 b , where
1  a  10 , and b is an integer
Ex3: Convert the given number to scientific
notation
a) 0.00000345
b) ( x 6 ) 7
b) 523,000,000
c) (4a 5 b 8 z 2 ) 3
4)Convert numbers from scientific
notation to standard notation
d) (2 x 4 b 6 c 6 ) 4
Ex4: Convert the given number to standard
notation
a) 3.07 x10 9
b) 7.56 x1010
5
e)
f)
y
y 4
5) Perform arithmetic operations using
numbers in scientific notation
Ex5: Perform the following calculations.
Express your answer using scientific
notation
a) (3.2 x10 6 )( 2.6 x1015 )
x 5  x 11
x 8
 a 4b 7
g)  5 8
 3c d



3
b) (2.0088x10 9 )  (5.4 x10 6 )
9
c) (87,000,000,000)(0.00002)
d) 0.000000064  16,000,000
6)Use scientific notation to solve applied
problems
Ex6: The speed of light is 1.86 x10 5 miles per
second. How far can light from the Sun
travel in 10 minutes?
Ex7: Burger King restaurants serve
7.7 x10 6 customers daily. How many
customers do Burger King restaurants serve
in one year?
Ex8: The pie chart shows the percent of U.S.
college students who received some type of
financial aid in 2003-2004. If there were
approximately 1.73 x10 7 U.S. college
students in 2003-2004, how many of them
received some type of financial aid?
10