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
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| More Examples
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Add / Subtract Radicals
• Recall how to collect like terms. The
method carries over to radicals.
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How to Divide Radicals
• If two radicals are defined and have the
same index then we can divide them.
 |
| Examples
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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:
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| 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
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|
|
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Irrational Numbers
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|
Set of Complex Numbers
|
Complex Numbers
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| Real Numbers
| Integers
| Whole Numbers
| Natural
Numbers
|
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Irrational Numbers
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| Examples of Arithmetic
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Applications of
Radical Equations |
| The Pythagorean Theorem
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| Special Right Triangles
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| Distance Formula
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| Equation of a Circle
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