10 1 radical expressions and graphs
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10.1 Radical Expressions and Graphs PowerPoint PPT Presentation


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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

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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 1 graph of a square root function

10.1 - Graph of a Square Root Function

(0, 0)


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 2 rational exponents2

10.2 Rational Exponents

  • Examples:


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 expressions2

10.3 Simplifying Radical Expressions

  • Example:

  • Example:


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 numbers1

10.7 Complex Numbers

  • Examples:


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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