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10.1 Radical Expressions and Graphs

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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  1. 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:

  2. 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:

  3. 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

  4. 10.1 - Graph of a Square Root Function (0, 0)

  5. 10.2 Rational Exponents • Definition: • All exponent rules apply to rational exponents.

  6. 10.2 Rational Exponents • Tempting but incorrect simplifications:

  7. 10.2 Rational Exponents • Examples:

  8. 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

  9. 10.3 Simplifying Radical Expressions • Product rule for radicals: • Quotient rule for radicals:

  10. 10.3 Simplifying Radical Expressions • Example: • Example:

  11. 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

  12. 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

  13. 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):

  14. 10.4 Adding and Subtracting Radical Expressions • We can add or subtract radicals using the distributive property. • Example:

  15. 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.

  16. 10.4 Adding and Subtracting Radical Expressions • Tempting but incorrect simplifications:

  17. 10.5 Multiplying and Dividing Radical Expressions • Use FOIL to multiply binomials involving radical expressions • Example:

  18. 10.5 Multiplying and Dividing Radical Expressions • Examples of Rationalizing the Denominator:

  19. 10.5 Multiplying and Dividing Radical Expressions • Using special product rule with radicals:

  20. 10.5 Multiplying and Dividing Radical Expressions • Using special product rule for simplifying a radical expression:

  21. 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:

  22. 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)

  23. 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.

  24. 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

  25. 10.7 Complex Numbers • Examples:

  26. 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)

  27. 10.7 Complex Numbers • Dividing by a complex number:

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