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Solving Equations with Rational Expressions

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: x2 = a
• Three cases:

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:

Higher Order Roots

• Start by Solving: x3 = a

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

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 a2 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 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

• 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