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

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