Solving Equations with Rational Expressions
How to Multiply
• If two radicals are defined and have the
same index then we can multiply them .
 |
Example
• Multiply the following, check using the
table.
|
Note
• It is important that the domain from the
original expression carry over to the
simplified version. So, the functions
 and
do not represent the same function
since they have different domains and
ranges. |
Simplifying and the Product
Rule
• Examples
 |
| More Examples
 |
Add / Subtract Radicals
• Recall how to collect like terms. The
method carries over to radicals.
 |
How to Divide Radicals
• If two radicals are defined and have the
same index then we can divide them.
 |
| Examples
 |
Differing Index
• If the two radicals to be multiplied or
divided have differing index, then we
need to fix that prior to doing the
arithmetic . Example:
 |
| More Examples
 |
How to Multiply (again)
• If there is more than one term in the
expression then we need to treat
multiplication as if we were multiplying
polynomials. That is we need to use the
distributive property. |
| Examples
 |
Rationalizing the Denominator
• Many times we want clear a radical
from the denominator of an expression .
We do this by multiplying the
expression by 1. |
| Example
 |
| Principal of Powers
If a=b then  for any exponent n
This is an "if- then" statement and the reverse
may not be true .That is :
If then a=b is not always true! |
In each of the following equations, the first has
been squared
To create the second. Solve each graphically to see what
effect squaring both sides of an equation has on the outcome.
 |
Caution!
• Raising both sides of an equation to an
even power may not produce an
equivalent equation . It is essential to
check your solutions. That is why the
NAG.
• Also, keep in mind that
 |
Solving Radical Equations
• Every equation will be investigated by
three methods: NAG!
• Numerical
• Algebraic
• Graphical |
| Solve the following NAG
 |
| Problems Involving Functions
If
 ,find a such that f(a)=5 |
Complex Numbers |
The Number i
• We define the number i such that
and
 |
| For Example
 |
Complex Numbers
• A complex number is any number that
can be written a + bi, where a and b are
real numbers . |
| The Set of Real Numbers
Real Numbers
| Real Numbers
| Integers
| Whole Numbers
| Natural Numbers
 |
|
|
|
Irrational Numbers
 |
|
Set of Complex Numbers
|
Complex Numbers
 |
| Real Numbers
| Integers
| Whole Numbers
| Natural
Numbers
|
|
|
|
Irrational Numbers
|
|
|
| Examples of Arithmetic
 |
Applications of
Radical Equations |
| The Pythagorean Theorem
 |
| Special Right Triangles
 |
| Distance Formula
 |
| Equation of a Circle
 |
Start solving your Algebra Problems
in next 5 minutes!
 |
 |
 |
|
Algebra Helper
Download (and optional CD)
Only $39.99
|
|
Click to Buy Now:
OR
|
|
|
 |
 |
 |
|
2Checkout.com is an authorized reseller
of goods provided by Sofmath
|
|
Attention: We are
currently running a special promotional offer
for Algebra-Answer.com visitors -- if you order
Algebra Helper by midnight of
November 2nd
you will pay only $39.99
instead of our regular price of $74.99 -- this is $35 in
savings ! In order to take advantage of this
offer, you need to order by clicking on one of
the buttons on the left, not through our regular
order page.
If you order now you will also receive 30 minute live session from tutor.com for a 1$!
|
You Will Learn Algebra Better - Guaranteed!
Just take a look how incredibly simple Algebra Helper is:
Step 1
: Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:
Step 2 :
Let Algebra Helper solve it:
Step 3 : Ask for an explanation for the steps you don't understand:
Algebra Helper can solve problems in all the following areas:
- simplification of algebraic expressions (operations
with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots
(radicals), absolute values)
- factoring and expanding expressions
- finding LCM and GCF
-
(simplifying, rationalizing complex denominators...)
- solving linear, quadratic and many other equations
and inequalities
(including basic logarithmic and exponential equations)
- solving a system of two and three linear equations
(including Cramer's rule)
- graphing curves (lines, parabolas, hyperbolas, circles,
ellipses, equation and inequality solutions)
- graphing general functions
- operations with functions (composition, inverse, range, domain...)
- simplifying logarithms
- basic geometry and trigonometry
(similarity, calculating trig functions, right triangle...)
- arithmetic and other pre-algebra topics
(ratios, proportions, measurements...)
ORDER NOW!
|
|
|
|
Algebra Helper
Download (and optional CD)
Only $39.99
|
|
Click to Buy Now:
OR
|
|
|
|
|
|
|
|
2Checkout.com is an authorized reseller
of goods provided by Sofmath
|
|
|
 |
| |
| "It
really helped me with my homework. I was
stuck on some problems and your software walked me
step by step through the process..." |
| C. Sievert, KY
| |
| |
 |
| |
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
|
Phone:
(512) 788-5675
Fax: (512) 519-1805
| | |
