Solving Equations with Rational Expressions

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Chapter 7

• Section 7
• Solving Equations with Rational Expressions

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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.

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Solve for x and Check Solution
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Solve for x Graphically
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).
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Solve Algebraically
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Solve for x and Check Solution on Calculator
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Your Turn Solve for x and Check Solution
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Chapter 8

• Section 1
• Introduction to Radicals

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Start by Solving x2 a

• Three casesa gt 0a 0a lt 0

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(No Transcript)
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Examples

• Solve the following graphically and
  algebraically

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Higher Order Roots

• Start by Solving x3 a

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Square Roots

• The number c is a square root of a if

Example So, -5 is a square root of 25
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Principal Square Root

• The principal square root is a non-negative
  number given by
• The negative square root is given by

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Note!

• For all real values of a

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Simplify The Square Root of a Square
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• For any real number aThe principal square
  root of a2 is the absolute value of a

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Simplify
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Higher Ordered Roots

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

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Examples, Use Graph or Table to Check
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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

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Definition of a Rational Exponent

• The nth root of of a is the same as raising a to
  the power of 1/n

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Definition of a Rational Exponent

• Also, given exists then

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Examples Rewrite in radical notation or in
rational exponents
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Simplify
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Examples

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

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Negative Exponents
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How to Multiply

• If two radicals are defined and have the same
  index then we can multiply them.

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Example

• Multiply the following, check using the table.

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Note

• It is important that the domain from the original
  expression carry over to the simplified version.
  So, the functionsdo not represent the same
  function since they have different domains and
  ranges.

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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.

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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
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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.

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Examples
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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.

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Example
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Principal of Powers
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In each of the following equations, the first has
been squared To create the second. Solve each
graphically to see whateffect squaring both
sides of an equation has on the outcome.
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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

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Solving Radical Equations

• Every equation will be investigated by three
  methods NAG!
• Numerical
• Algebraic
• Graphical

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Solve the following NAG
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Problems Involving Functions
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• Complex Numbers

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The Number i

• We define the number i such that

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For Example
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Complex Numbers

• A complex number is any number that can be
  written a bi, where a and b are real numbers.

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The Set of Real Numbers
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Set of Complex Numbers
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Examples of Arithmetic
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• Applications ofRadical Equations

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The Pythagorean Theorem
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Special Right Triangles
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Distance Formula
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Equation of a Circle