
Course Goals
We will learn to:
Solve linear and quadratic equations
and inequalities
Graph lines, circles, and parabolas
Graph inequalities
Apply linear, quadratic, and other
models to solve problems
Use exp onential and logarithmic
functions
Count 
Required Materials
You will need
the book
Jerome E. Kaufmann, College Algebra , 6/e
a calculator
At minimum, your calculator should have
arbitrary exponents
or
exponentials
logarithms
factorials
permutations
combinations

Your Grade
Weekly homework — 15%
Problems listed in syllabus
Each week’s problems due the
following Wednesday
6–8 problems graded
Three hour exams — 15% each = 45%
Final exam — 40% 
Classroom Expectations
Attend class
Not mandatory
No attendance points or
participation points
A good deal of lecture material will
not be in the slides
Do not make yourself a distraction
No loud conversations
No newspaper rustling
No stinky or noisy food
Most importantly, ask questions! 
Outline
Review of Basic Concepts
The Real Numbers
Fractions
Integer Exponents
Positive Integer Exponents
Zero as an Exponent
Negative Integer Exponents
Scientific Notation
Radicals
Definition and Notation
Simplest Radical Form
Fractional Exponents 
The Real Numbers
Integers

The Real Numbers
RationalsThe
rationals are the set of all possible quotients formed using
integers
A rational number can be written as
where p and q are
integers with
A rational number can also be
written in decimal form, and will
either terminate
or repeat

The Real Numbers
Completing the Reals
Irrational numbers cannot be ex pressed as fractions . This means
they have decimal representations that neither terminate nor
repeat.
The reals are difficult to define. For now, let’s just say the reals
comprise the rationals and the irrationals. 
Fractions
Simplifying Fractions
We can simplify fractions by canceling the same thing
from the
numerator and the denominator .

Fractions
Arithmetic with Fractions
To add or subtract fractions , get a common
denominator, and then
add or subtract the numerators.

Fractions
Arithmetic with Fractions
To multiply fractions, multiply the numerators and
denominators
separately. Cancel first if possible.

Fractions
Arithmetic with Fractions
To divide fractions, multiply by the reciprocal of
the divisor .

Positive Integer Exponents
Terminology and Notation
Integer exponents denote repeated multiplication 
The base is
repeatedly multiplied by itself
The exponent
indicates the number of factors
Nitpick
An exponent does not indicate the “number of times the base is
multiplied by itself” 

Positive Integer Exponents
First Definition of b^{n}
Definition
For any positive integer n and any real number b,


Positive Integer Exponents
Properties of Exponents
Properties of Exponents
For any positive integers m and n, and for any real numbers a
and
b,


Zero as an Exponent
What Should a Zero Exponent Mean?
Zero exponents should satisfy the same properties as
positive
exponents. Let m = 0 in Property 1:
So, unless b is zero, b^{0} must be 1.
Definition
For any nonzero real number b,


Negative Integer Exponents
What Should a Negative Exponent Mean?
Again, we want negative exponents to satisfy the same
properties
as positive exponents. Set m = −n in Property 1:
So b^{n} must be the reciprocal of b^{n}.
Definition
For any positive integer n and any nonzero real number b,


Negative Integer Exponents
Restating the Properties of Exponents
Properties of Exponents
For any integers m and n, and for any real numbers a and b,


Scientific Notation
Scientific Notation
Definition
A real number written in the form
or
with 1 ≤n < 10 and k an integer, is in
scientific notation. 
To write a number in scientific notation,
Move the decimal point to the immediate right of the first
nonzero number—this produces n
Count the number of places the decimal point moved—this
gives you k
If you moved the decimal point to the left, k is positive; if you
moved it right, k is negative. 
Examples
Examples
Evaluate
Evaluate
Simplify
. Use only nonnegative exponents.
Express 0.000000078 in scientific
notation.
Use scientific notation to evaluate

Definition and Notation
Definition of
The book’s definition is wrong!
Definition
For any nonnegative real number a,
if and only if

Note: Even though
is 2, not −2.
For any real number b, if
exists, it is nonnegative. 

Definition and Notation
Definition of
Definition
For any positive integer n and nonnegative real number a,
if and only if
If n is odd, this holds for any (possibly negative) real number
a. 
Examples

Simplest Radical Form
Properties of Radicals
Notice that there is no property for
A sum of radicals
generally cannot be simplified.
Also notice how similar these properties are to the properties of
exponents. . 
Simplest Radical Form
Simplest Radical Form
An example of the last condition is
is a factor of 75, and
25 is a perfect square.

Simplest Radical Form
Rationalizing the Denominator
To clear a radical from the denominator, multiply the numerator and
the denominator by a radical that will allow you to remove the
radical sign.
In case there is a sum (or difference) of radicals
in the
denominator, say multiply by the
conjugate 
Examples
Examples
Express
in simplest radical form.
Express in simplest radical form.
Rationalize the denominator and simplify
Rationalize the denominator and simplify

Roots are Exponents!
The nth root of b
Recall that . Suppose m = 1/n. Then
So is the nth root of b.
Definition
For any positive integer n and any real number b, if
exists, then


Roots are Exponents!
Definition of
Because we have
and
Definition
For any positive integer n and any real number b, if m/n is a
rational number written in lowest terms, and if
exists, then


Examples
Examples
Evaluate
Evaluate
Evaluate
Express
in simplest radical form.
Simplify
into one radical.
