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.
![](./articles_imgs/5827/solvin55.gif) |
Solve for x and Check
Solution![](./articles_imgs/5827/solvin56.gif)
|
Your Turn: Solve for x and
Check Solution
![](./articles_imgs/5827/solvin57.gif) |
Solve for x and Check
Solution![](./articles_imgs/5827/solvin58.gif)
|
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: x2 = a
• Three cases:
![](./articles_imgs/5827/solvin60.gif) |
Define 5. The solution of x2=a
are called "square root of a" • In the
case a<0 ,the equation x2=a has no real solutions.
• In the case a=0 ,the equation x2=a
has one real solution ,namely x=0
• In the case a>0, the equation x2=a
has two real solution , The
notation calls for the positive
square root of a ,that is ,the positive solution of x2=a.
The notation calls for the negative
square root of a , that is ,
the negative equation solution of x2=a. |
Examples
• Solve the following graphically and
algebraically:
![](./articles_imgs/5827/solvin61.gif) |
Higher Order Roots
• Start by Solving: x3 = a
![](./articles_imgs/5827/solvin31.gif) Figure
2.The graph of y=x3 inter-
sect the graph of y=a in exactly one
place. |
Square Roots
• The number c is a square root of a if
![](./articles_imgs/5827/solvin63.gif)
Example:![](./articles_imgs/5827/solvin64.gif)
So, -5 is a square root of 25 |
Principal Square Root
• The principal square root is a nonnegative
number given by:
![](./articles_imgs/5827/solvin65.gif)
• The negative square root is given by:
![](./articles_imgs/5827/solvin66.gif) |
Note!
• For all real values of a
![](./articles_imgs/5827/solvin18.gif) |
Simplify The Square Root
of a Square![](./articles_imgs/5827/solvin19.gif)
|
• For any real number a
![](./articles_imgs/5827/solvin20.gif)
The principal square root of a2 is the
absolute value of a |
Simplify
![](./articles_imgs/5827/solvin70.gif) |
Higher Ordered Roots
• The value c the nth root of a if
![](./articles_imgs/5827/solvin21.gif)
• The nth root of a number is denoted
![](./articles_imgs/5827/solvin22.gif)
|
Examples, Use Graph or
Table to Check
![](./articles_imgs/5827/solvin73.gif) |
The nth root of an
• 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 nth root of of a is the same as
raising a to the power of 1/n
![](./articles_imgs/5827/solvin77.gif) |
• Also, given exists then
![](./articles_imgs/5827/solvin79.gif) |
Examples: Rewrite in radical
notation or in rational
exponents
![](./articles_imgs/5827/solvin80.gif) |
Simplify
![](./articles_imgs/5827/solvin81.gif)
![](./articles_imgs/5827/solvin82.gif)
|
Examples
• Use the table (where possible) to
determine if the following simplifications
are correct
![](./articles_imgs/5827/solvin83.gif) |
Negative Exponents
also
![](./articles_imgs/5827/solvin85.gif)
|