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Powers, Roots, and Radicals

Powers, Roots, and Radicals

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Powers, Roots, and Radicals

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  1. Powers, Roots, and Radicals Algebra 2 Chapter 7 Mr. Hardy

  2. DO NOW • Evaluate the expression • 1) √9 • 2) -√121 • 3) (√25)2 • Solve each equation • 4) x2 = 49 • 5) (x – 1)2 = 64

  3. Chapter 7 Overview • Powers, roots, and radicals • How to use rational exponents and nth roots of numbers • How to perform operations with and find inverses of functions • How to graph radical functions and solve radical equations

  4. Chapter 7.1 Lesson Opener • You will havetwominutes to complete the lesson opener in your notebook. • Evaluate • (2)(2)(2)(2) • (-2)4 • (-2)(-2)(-2) • (2)(2)(2)

  5. nth roots • For an integer n greater than 1, if bn = a, then b is an nth root of a. • An nth root of a is written: • where nis the index of the radical

  6. DO NOW • Page 400 • #1 – 13

  7. Notice • An nth root can also be written as a power of a. • Suppose:

  8. THEREFORE In General Where n is greater than one (positive)

  9. Real nth Roots

  10. Finding nth Roots • Find the indicated real nth root(s) of a • n = 2, a = 16 • n is even, and a > 0; therefore, a has two real nth roots • √16 • √16 = ±4

  11. Example 2 • n = 3, a = -1 • n is odd; therefore, a has one real nth root • Hint: What to the power of 3 is -1?

  12. Example 3 • n = 4, a = -16 • n is even, but a is negative. How many real nth roots?

  13. Try These • n = 3, a = 125 • n = 3, a = 0 • n = 2, a = 49 • 5 • 0 • 7

  14. Rational Exponents • Keep in mind: Rational exponents do not have to be in the form of 1/n. Other rational numbers can be used as exponents as well!

  15. Example 1

  16. Example 2

  17. Try These • 9 • 7 • 1/27

  18. Solving Equations • Example 1 • - 5x2 = -30 • x2 = 6 • x = ±√6 • Use a calculator to round the result • x ≈ ±2.45

  19. Example 2 • (x + 4)3 = 27 • x + 4 = 3 (Take the cube root of both sides) • x = -1

  20. Try These • x4 = 87 • 2x3 = 92 • (x – 1)5 = 12 • ±3.05 • 3.58 • 2.64

  21. Your Turn Classwork Homework • Complete Practice B worksheet • Chapter 7.1, page 404 #14-60 even, 65-66 all

  22. Challenge • Rationalize each denominator, and express the fraction in simplest form

  23. Do Now 3-15-13 • Homework Quiz • Chapter 6.8 • #24, 26, 28 • Chapter 7.1 • #16, 28, 40, 56

  24. Recall • Rewrite the Expression • 1) • 2) • 3) • Solve the equation. Round if necessary • 2x4 = 35 • 5x3 + 10 = 961

  25. Section 7.2 Opener • Evaluate the expression • 1) 42 43 • 2) (22)3 • 3) 34/32 • 4) 2-3 • 5) (2  3)4

  26. Properties of (Rational) Exponents

  27. Product and Quotient Property

  28. Using Properties of Rational Exponents • Example 1 • Same bases- ADD the EXPONENTS

  29. Example 2 • MULTIPLY the EXPONENTS • (21)(x2) = 2x2

  30. Example 3

  31. Try These • x1/2 • 256 • y1/3 • 4

  32. Simplest Form • For a radical to be in simplest form, you must apply the properties of radicals, remove any perfect nth powers (other than 1) and rationalize any denominators. • Two radical expressions are like radicals if they have the same index and the same radicand.

  33. Examples • √2 * √8 • = √2*8 • =√16 • =4

  34. Try These

  35. Writing Radicals and Variable Expression in Simplest Form • Example 1 • Factor out the perfect fourth root • Use the Product Property

  36. Example 2 • Make the denominator a perfect cube (in order to rationalize

  37. Do Now Looseleaf Silently Agree or Disagree with the statement that follows and support it with specific and accurate facts that include explanations/proofs, sentences, number statements, and worked out mathematical solutions. The expression can be rewritten as

  38. Recall • Simplify

  39. Recall: nth roots w/ variables  RECALL from Chapter 5 FACTOR out the PERFECT SQUARE

  40. Example 2

  41. Try These

  42. Application Use what you know to find a radical expression for the perimeter of the triangle. Simplify the expression. 4ft 1ft 2ft 2ft

  43. The function f(x) = 70x3/4… • …models the number of calories per day, f(x), a person needs to maintain life in terms of that person's weight, x, in kilograms. (1 kilogram is approximately 2.2 pounds.) Use this model and a calculator to solve.Round answers to the nearest calorie. • How many calories per day does a person who weighs 80 kilograms (approximately 176 pounds) need to maintain life?70 kilograms (approximately 154 pounds)?

  44. Operating Radicals

  45. Challenge • Solve the equation

  46. Chapter 7.3Operations on Functions • Addition: h(x) = f(x) + g(x) • Subtraction: h(x) = f(x) – g(x) • Multiplication: h(x) = f(x)*g(x) or f(x)g(x) • Division: h(x) = f(x)/g(x) or f(x) ÷ g(x) • Composition h(x) = f(g(x)) OR g(f(x))

  47. Power Functions • y = axb • Where a is a real number and b is a rational number • Note, when b is a positive integer, a power function is simply a type of polynomial function

  48. Note The rules for thedomainof functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g. For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

  49. Addition Example • f(x) = 3x + 8; g(x) = 2x – 12 • (f + g)(x) = f(x) + g(x) = (3x + 8) + (2x – 12) • What like terms do we have? Combine them and then we have our new function. • (f + g)(x) = 5x - 4

  50. Subtraction Example • f(x) = 5x2 – 4x; g(x) = 5x + 1 • (f – g)(x) = f(x) – g(x) = (5x2 – 4x) – (5x + 1) • What like terms do we have? Combine them and then we have our new function. • Be careful that you know you are subtracting the whole function g(x) • (f – g)(x) = 5x2 – 9x – 1