10.1 Radical Expressions and Graphs

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10.1 Radical Expressions and Graphs

• is the positive square root of a, andis
  the negative square root of a because
• If a is a positive number that is not a perfect
  square then the square root of a is irrational.
• If a is a negative number then square root of a
  is not a real number.
• For any real number a

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10.1 Radical Expressions and Graphs

• The nth root of a is the nth root of a.
  It is a number whose nth power equals a, so
  
• n is the index or order of the radical
• Example

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10.1 Radical Expressions and Graphs

• The nth root of nth powers
• If n is even, then
• If n is odd, then
• The nth root of a negative number
• If n is even, then the nth root is not a real
  number
• If n is odd, then the nth root is negative

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10.1 - Graph of a Square Root Function
(0, 0)
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10.2 Rational Exponents

• Definition
• All exponent rules apply to rational exponents.

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10.2 Rational Exponents

• Tempting but incorrect simplifications

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10.2 Rational Exponents

• Examples

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10.3 Simplifying Radical Expressions

• Review Expressions vs. Equations
• Expressions
• No equal sign
• Simplify (dont solve)
• Cancel factors of the entire top and bottom of a
  fraction
• Equations
• Equal sign
• Solve (dont simplify)
• Get variable by itself on one side of the
  equation by multiplying/adding the same thing on
  both sides

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10.3 Simplifying Radical Expressions

• Product rule for radicals
• Quotient rule for radicals

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10.3 Simplifying Radical Expressions

• Example
• Example

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10.3 Simplifying Radical Expressions

• Simplified Form of a Radical
• All radicals that can be reduced are reduced
• There are no fractions under the radical.
• There are no radicals in the denominator
• Exponents under the radical have no common factor
  with the index of the radical

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10.3 Simplifying Radical Expressions

• Pythagorean Theorem In a right triangle, with
  the hypotenuse of length c and legs of lengths a
  and b, it follows that c2 a2 b2
• Pythagorean triples (integer triples that satisfy
  the Pythagorean theorem) 3, 4, 5, 5, 12,
  13, 8, 15, 17

c
a
90?
b
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10.3 Simplifying Radical Expressions

• Distance Formula The distance between 2 points
  (x1, y1) and (x2,y2) is given by the formula
  (from the Pythagorean theorem)

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10.4 Adding and Subtracting Radical Expressions

• We can add or subtract radicals using the
  distributive property.
• Example

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10.4 Adding and Subtracting Radical Expressions

• Like Radicals (similar to like terms) are
  terms that have multiples of the same root of the
  same number. Only like radicals can be combined.

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10.4 Adding and Subtracting Radical Expressions

• Tempting but incorrect simplifications

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10.5 Multiplying and Dividing Radical Expressions

• Use FOIL to multiply binomials involving radical
  expressions
• Example

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10.5 Multiplying and Dividing Radical Expressions

• Examples of Rationalizing the Denominator

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10.5 Multiplying and Dividing Radical Expressions

• Using special product rule with radicals

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10.5 Multiplying and Dividing Radical Expressions

• Using special product rule for simplifying a
  radical expression

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10.6 Solving Equations with Radicals

• Squaring property of equality If both sides of
  an equation are squared, the original solution(s)
  of the equation still work plus you may add
  some new solutions.
• Example

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10.6 Solving Equations with Radicals

• Solving an equation with radicals
• Isolate the radical (or at least one of the
  radicals if there are more than one).
• Square both sides
• Combine like terms
• Repeat steps 1-3 until no radicals are remaining
• Solve the equation
• Check all solutions with the original equation
  (some may not work)

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10.6 Solving Equations with Radicals

• ExampleAdd 1 to both sidesSquare both
  sidesSubtract 3x 7So x -2 and x 3,
  but only x 3 makes the original equation equal.

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10.7 Complex Numbers

• Definition
• Complex Number a number of the form a bi where
  a and b are real numbers
• Adding/subtracting add (or subtract) the real
  parts and the imaginary parts
• Multiplying use FOIL

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10.7 Complex Numbers

• Examples

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10.7 Complex Numbers

• Complex Conjugate of a bi a
  bimultiplying by the conjugate
• The conjugate can be used to do division(similar
  to rationalizing the denominator)

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10.7 Complex Numbers

• Dividing by a complex number