Solving Equations by Clearing Fractions
• Recall how to solve an equation
containing fractions. We found the LCD
of all denominators and “cleared
fractions” by multiplying both sides of
the equation by the LCD.

Solve for x and Check
Solution 
Your Turn: Solve for x and
Check Solution

Solve for x and Check
Solution 
Solve for x Graphically
find all x such
that f(x)=12
a. Sketch the graph of f on graph paper. Label the
zeros of f with their
coordinates and the asymptotes of f with their equations.
b. Add the graph of y = 12 to your plot and estimate the coordinates of
where the graph of f intersects the graph of y = 12.
c. Use the intersect utility on your calculator to find better
approximations of the points where the graphs of f and y = 12
intersect.
d. Solve the equation f(x) = 2 algebraically and compare your solutions
to those found in part (c). 
Your Turn
• Page 677: 1 and 2 
Chapter 8
Section 1
Introduction to Radicals 
Start by Solving: x^{2} = a
• Three cases:

Define 5. The solution of x^{2}=a
are called "square root of a" • In the
case a<0 ,the equation x^{2}=a has no real solutions.
• In the case a=0 ,the equation x^{2}=a
has one real solution ,namely x=0
• In the case a>0, the equation x^{2}=a
has two real solution , The
notation calls for the positive
square root of a ,that is ,the positive solution of x^{2}=a.
The notation calls for the negative
square root of a , that is ,
the negative equation solution of x^{2}=a. 
Examples
• Solve the following graphically and
algebraically:

Higher Order Roots
• Start by Solving: x^{3} = a
Figure
2.The graph of y=x^{3} inter
sect the graph of y=a in exactly one
place. 
Square Roots
• The number c is a square root of a if
Example:
So, 5 is a square root of 25 
Principal Square Root
• The principal square root is a nonnegative
number given by:
• The negative square root is given by:

Note!
• For all real values of a

Simplify The Square Root
of a Square 
• For any real number a
The principal square root of a^{2} is the
absolute value of a 
Simplify

Higher Ordered Roots
• The value c the nth root of a if
• The nth root of a number is denoted

Examples, Use Graph or
Table to Check

The nth root of a^{n}
• To simplify where a is any real
• The value of when n is even
• The value of when n is odd 
Definition of a Rational
Exponent
• The n^{th} root of of a is the same as
raising a to the power of 1/n

• Also, given exists then

Examples: Rewrite in radical
notation or in rational
exponents

Simplify

Examples
• Use the table (where possible) to
determine if the following simplifications
are correct

Negative Exponents
also
