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10.1 Radical Expressions and Graphs. is the positive square root of a, and is 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.

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10 1 radical expressions and graphs
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:
10 1 radical expressions and graphs1
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:
10 1 radical expressions and graphs2
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
10 2 rational exponents
10.2 Rational Exponents
  • Definition:
  • All exponent rules apply to rational exponents.
10 2 rational exponents1
10.2 Rational Exponents
  • Tempting but incorrect simplifications:
10 3 simplifying radical expressions
10.3 Simplifying Radical Expressions
  • Review: Expressions vs. Equations:
    • Expressions
      • No equal sign
      • Simplify (don’t solve)
      • Cancel factors of the entire top and bottom of a fraction
    • Equations
      • Equal sign
      • Solve (don’t simplify)
      • Get variable by itself on one side of the equation by multiplying/adding the same thing on both sides
10 3 simplifying radical expressions1
10.3 Simplifying Radical Expressions
  • Product rule for radicals:
  • Quotient rule for radicals:
10 3 simplifying radical expressions3
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
10 3 simplifying radical expressions4
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

10 3 simplifying radical expressions5
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):
10 4 adding and subtracting radical expressions
10.4 Adding and Subtracting Radical Expressions
  • We can add or subtract radicals using the distributive property.
  • Example:
10 4 adding and subtracting radical expressions1
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.
10 4 adding and subtracting radical expressions2
10.4 Adding and Subtracting Radical Expressions
  • Tempting but incorrect simplifications:
10 5 multiplying and dividing radical expressions
10.5 Multiplying and Dividing Radical Expressions
  • Use FOIL to multiply binomials involving radical expressions
  • Example:
10 5 multiplying and dividing radical expressions1
10.5 Multiplying and Dividing Radical Expressions
  • Examples of Rationalizing the Denominator:
10 5 multiplying and dividing radical expressions2
10.5 Multiplying and Dividing Radical Expressions
  • Using special product rule with radicals:
10 5 multiplying and dividing radical expressions3
10.5 Multiplying and Dividing Radical Expressions
  • Using special product rule for simplifying a radical expression:
10 6 solving equations with radicals
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:
10 6 solving equations with radicals1
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)
10 6 solving equations with radicals2
10.6 Solving Equations with Radicals
  • Example:Add 1 to both sides:Square both sides:Subtract 3x + 7:So x = -2 and x = 3, but only x = 3 makes the original equation equal.
10 7 complex numbers
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
10 7 complex numbers2
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)
10 7 complex numbers3
10.7 Complex Numbers
  • Dividing by a complex number:
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