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Chapter 6

Chapter 6. Irrational and Complex Numbers. Section 6-1. Roots of Real Numbers. Square Root. A square root of a number b is a solution of the equation x 2 = b. Every positive number b has two square roots, denoted √b and -√b. Principal Square Root.

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Chapter 6

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  1. Chapter 6 Irrational and Complex Numbers

  2. Section 6-1 Roots of Real Numbers

  3. Square Root • A square root of a number b is a solution of the equation x2 = b. Every positive number b has two square roots, denoted √b and -√b.

  4. Principal Square Root • The positive square root of b is the principal square root • The principal square root of 25 is 5

  5. Examples – Square Root • Simplify • x2 = 9 • x2 + 4 = 0 • 5x2 = 15

  6. Cube Root • A cube root of b is a solution of the equation x3 = b.

  7. Examples – Cube Root • Simplify • 3√8 • 3√27 • 3√106 • 3√a9

  8. nth root • is the solution of xn = b • If n is even, there could be two, one or no nth root • If n is odd, there is exactly one nth root

  9. Examples – nth root • Simplify • 4√81 • 5√32 • 5√-32 • 6√-1

  10. Radical • The symbol n√b is called a radical • Each symbol has a name • n = index • √ = radical • b = radicand

  11. Section 6-2 Properties of Radicals

  12. Product and Quotient Properties of Radicals 1. n√ab = n√a · n√b 2. n√a÷b = n√a ÷ n√b

  13. Examples • Simplify • 3√25 · 3√10 • 3√(81/8) • √2a2b • √36w3

  14. Rationalizing the Denominator • Create a perfect square, cube or other power in the denominator in order to simplify the answer without a radical in the denominator

  15. Examples • Simplify • √(5/3) • 4 3√c

  16. Theorems 1.If each radical represents a real number, then nq√b = n√(q√b). 2. If n√b represents a real number, then n√bm = (n√b)m

  17. Examples • Give the decimal approximation to the nearest hundredth. • 4√100 • 3√1702

  18. Section 6-3 Sums of Radicals

  19. Like Radicals • Two radicals with the same index and same radicand • You add and subtract like radicals in the same way you combine like terms

  20. Examples • Simplify • √8 + √98 • 3√81 - 3√24 • √32/3 + √2/3

  21. Examples • Simplify • √12x5 - x√3x3 + 5x2√3x • Answer • 6x2√3x

  22. Section 6-4 Binomials Containing Radicals

  23. Multiplying Binomials • You multiply binomials with radicals just like you would multiply any binomials. • Use the FOIL method to multiply binomials

  24. Examples • Simplify • (4 + √7)(3 + 2√7) • Answer • 26 + 11√7

  25. Conjugate • Expressions of the form a√b + c√d and a√b - c√d • Conjugates can be used to rationalize denominators

  26. Example - Conjugate • Simplify 3 + √5 3 - √5 • Answer 7 + 3√5 2

  27. Example - Conjugate • Simplify • 1 4 - √15 • Answer • 4 + √15

  28. Section 6-5 Equations Containing Radicals

  29. Radical Equation • An equation which contains a radical with a variable in the radicand.

  30. Solving a Radical Equation • First isolate the radical term on one side of the equation

  31. Solving a Radical Equation - Continued • If the radical term is a square root, square both sides • If the radical term is a cube root, cube both sides

  32. Example 1 • Solve • Answer • X = 5

  33. Example 2 • Solve • Answer • X = 9

  34. Example 3 • Solve • Answer • X = 2/9

  35. Section 6-6 Rational and Irrational Numbers

  36. Completeness Property of Real Numbers • Every real number has a decimal representation, and every decimal represents a real number

  37. Remember… • A rational number is any number that can be expressed as the ratio or quotient of two integers

  38. Decimal Representation • Every rational number can be represented by a terminating decimal or a repeating decimal

  39. Example 1 • Write each terminating decimal as a fraction in lowest terms. • 2.571 • 0.0036

  40. Example 2 • Write each repeating decimal as a fraction in lowest terms. • 0.32727… • 1.89189189…

  41. Remember… • An irrational number is a real number that is not rational

  42. Decimal Representation • Every irrational number is represented by an infinite and nonrepeating decimal • Every infinite and nonrepeating decimal represents an irrational number

  43. Example 3 • Classify each number as either rational or irrational √2 √4/9 2.0303… 2.030030003…

  44. Section 6-7 The Imaginary Number i

  45. Definition i = √-1 and i2 = -1

  46. Definition • If r is a positive real number, then √-r = i√r

  47. Example 1 • Simplify • √-5 • √-25 • √-50

  48. Combining imaginary Numbers • Combine the same way you combine like terms • √-16 - √-49 • i√2 + 3i√2

  49. Multiply - Example • Simplify • √-4 ▪ √-25 • i√2 ▪ i√3

  50. Divide - Example • Simplify • 2 3i • 6 √-2

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