English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


College Algebra and Probability

  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
Rationals

The 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 bn

 

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, b0 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 bn.

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

 

Properties of Radicals
If
and are real numbers, then

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

 

Definition
An expression is said to be in simplest radical form if:
No fraction appears within a radical sign.
No radical appears in the denominator.
No radicand contains a factor that is a perfect power of the
index.

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.

Prev Next