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Intermediate Algebra Chapter 7 - Gay

Intermediate Algebra Chapter 7 - Gay. Radical Expressions. Oprah Winfrey. “Although there may be tragedy in your life, there’s always to possibility to triumph. It doesn’t matter who you are, where you come from. The ability to triumph begins with you. Always.”.

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Intermediate Algebra Chapter 7 - Gay

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  1. Intermediate AlgebraChapter 7 - Gay • Radical Expressions

  2. Oprah Winfrey • “Although there may be tragedy in your life, there’s always to possibility to triumph. It doesn’t matter who you are, where you come from. The ability to triumph begins with you. Always.”

  3. Angela Davis – U.S. political activist-1987 – Spellman college • “Radical simply means grasping things at the root.”

  4. Intermediate Algebra 7.1 • Radicals

  5. Objective • Find the nth root of a number

  6. Definition of nth root • For any real numbers a and b and any integer n>1, a is a nth root of b if and only if

  7. Principal nth root • Even roots • Principal nth root of b is the nonnegative nth root of b. • Represented by

  8. Graphs determine domain & range

  9. Graphs – determine domain & range

  10. Calculator keys

  11. If b is any real number • For even integers

  12. If b is any real number • For odd integers n

  13. Objectives • 1. Find the nth root of a number • 2. Approximate roots using calculator. • 3. Graph radical functions • 4. Determine domain and range of radical functions. • 5. Simplify radical expressions.

  14. Intermediate Algebra 7.2 • Rational Exponents

  15. Rational Exponent – numerator of 1 • For any real number b for which the nth roof of b is defined and any integer n>1

  16. Definition of

  17. Problem

  18. Negative exponents

  19. Rule & example

  20. Althea Gibson – tennis player • “No matter what accomplishments you make, someone helped you.”

  21. Intermediate Algebra 8.3 • Properties • of • Rational Exponents

  22. Properties of exponents

  23. Procedure: Reduce the Index • 1. Write the radical in exponential form • 2. Reduce exponent to lowest terms. • 3. Write the exponential expression as a radical.

  24. Objectives: • 1. Evaluate rational exponents. • 2. Write radicals as expressions raised to rational exponents. • 3. Simplify expressions with rational number exponents using the rules of exponents. • 4. Simplify radical expressions

  25. Thomas Edison • “I am not discouraged, because every wrong attempt discarded is another step forward.”

  26. Intermediate Algebra 7.3 • The Product Rule • for • Radicals

  27. Product Rule for Radicals • For all real numbers a and b for which the operations are defined • The product of the radicals is the radical of the product.

  28. Simplifying a RadicalCondition 1 • The radicand of a simplified n-th root radical must not contain a perfect n-th power factor.

  29. Using product rule to simplify • 1. Write the radicand as a product of the greatest possible perfect nth power and a number that has no perfect nth power factors. • 2. Use product rule • 3. Find the nth root of perfect nth power radicand. • 4. Do all necessary simplifications

  30. Sample problem

  31. Sample Problem

  32. Winston Churchill • “I am an optimist.”

  33. Intermediate Algebra 7.5 • The Quotient Rule • for • Radicals

  34. Quotient Rule for Radicals • For all real numbers a and b for which the operations are defined. • The radical of a quotient is the quotient of the radical.

  35. Simplifying a radical: condition 2 • The radicand of a simplified radical must not contain a fraction

  36. Simplifying a radical – condition 3 • A simplified radical must not contain a radical in the denominator.

  37. Rationalizing the denominator • Square Roots • 1. Multiply both the numerator and denominator by the same square root as appears in the denominator. • 2. Simplify.

  38. Sample problem

  39. Rationalizing a denominator containing a higher-order radical. • Multiply the numerator and denominator by the expression that will make the radicand of the denominator a perfect nth power.

  40. Example problem

  41. Stanislaw J. Lec • “He who limps is still walking.”

  42. Intermediate Algebra 8.6 • Operations • with • Radicals

  43. Objective • Add or subtract like radicals

  44. Definition: Like Radicals • Are radical expressions • * with identical radicands • and • * Identical indexes.

  45. Procedure – Adding like radicals • Simplify all radicals first. • To add or subtract like radicals, add or subtract the coefficients and keep the radicals the same.

  46. Procedure- multiplication with radicals • Simplify all radicals first • Use Product Rule • Use distributive property • Use FOIL if needed

  47. Conjugates • A+B and A-B are called conjugates of each other. • Examples:

  48. Rationalizing a binomial denominator with radicals • Multiply the numerator and denominator by the conjugate of the denominator. • Combine and Simplify • Denominator cannot be radical

  49. Rationalizing a binomial numerator with radicals • Multiply the numerator and denominator by the conjugate of the numerator. • Combine and Simplify • Denominator cannot be radical

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