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Lesson 10-1: The Pythagorean Theorem

Lesson 10-1: The Pythagorean Theorem. Simplifying Radicals . Lesson 10-2. Essential Understanding . A radical is any number with a square root. You can simplify a radical expressions using multiplication and division. MULIPLICATION PROPERTY OF SQUARE ROOTS: √32 = √16 ∙ √2 = 4√2.

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Lesson 10-1: The Pythagorean Theorem

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  1. Lesson 10-1: The Pythagorean Theorem

  2. Simplifying Radicals Lesson 10-2

  3. Essential Understanding • A radical is any number with a square root. • You can simplify a radical expressions using multiplication and division. • MULIPLICATION PROPERTY OF SQUARE ROOTS: • √32 = √16 ∙ √2 = 4√2

  4. Problem 1 • What is the simplified form of √160? • Ask “what perfect square goes into 160?” • 160 = 2 ∙ 80 No perfect square • 160 = 4 ∙ 40 Yes perfect square • 160 = 16 ∙ 10 Yes perfect square • √160 = √16 ∙ √10 = 4√10

  5. Got it? 1 • Simplify: √72

  6. Problem 2 • Simplify: √54n7 • √9 ∙ 6 ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n • √9 ∙ √6 ∙ √n2∙ √n2∙ √n2∙ √n • 3 ∙ √6 ∙ n ∙ n ∙ n ∙ √n • 3n3∙√6n • 3n3√6n

  7. Got it? 2 • Simplify: -m √80m9

  8. Problem 3 • Simplify: 2√7t ∙ 3√14t2 • 2 ∙ 3 ∙ √ 7t ∙ 14t2 • 6 ∙ √98t3 • 6 ∙ √49t2 ∙ 2t • 6 ∙ 7t ∙ √2t • 42t√2t

  9. Division Property of Square Root √ 36 49 √ 144 36

  10. Problem 5 √ 8x3 50x 8x3 = √4  2  x2 x 50x = √25  2  x 2 x 5 2x 5

  11. Rationalizing the Denominator • It’s okay to have a square root in the numerator, but not the denominator. It’s not simplified enough if you keep a square root in the denominator. √3 √7 √3 √7 √7 √7 √21 √49 √21 7 Really equals 1

  12. Operations With Radical Expressions Lesson 10-3

  13. Combining “like” radicals 3√5 and 7√5 have the same radicand. Radicand = number under the square root. -2√9 and 4√3 do not have the same radicand. If two or more numbers have the same radicand, then we can combine them together.

  14. Problem 1 What is the simplified form of 2√11 + 5√11? 2√11 + 5√11 We could break it down even more… (√11 + √11) + (√11 + √11 + √11 + √11 + √11) How many √11’s do we have altogether? 7 √11 2√11 + 5√11 = 7 √11

  15. What is the simplified form of √3 - 5√3? √3 - 5√3 1√3 - 5√3 (1 – 5)√3 -4√3 Got it? 1. 7√2 - 8√2 2. 5√5 + 6√5

  16. Problem 2: What if they don’t “look like they can be simplified? 5√3 - √12 Simplify √12 to see if there is a perfect square. √12 = √4 ∙ 3 = 2√3 So we have 5√3 - 2√3. 5√3 - 2√3 = 3√3

  17. Got it? 2 • 4 √7 + 2 √28 • 5 √32 - 4 √18

  18. Problem 3: Using the Distributive Property 10(6 + 3) = 10(6) + 10(3) = 60 + 30 = 90 In the same way… √10(√6 + 3) Use the Distributive Property (√10 ∙ √6) + (√10 ∙ 3) √60 + 3√10 Break down 60 to find a perfect square. √4 ∙ √15 + 3√10 2 √15 + 3 √10 Can we simplify even more?

  19. Problem 3: (√6 - 2 √3)(√6 + √3) (√6-2 √3)(√6 + √3) Carefully FOIL (√6)(√6)+ (- 2 √3)(√6) + (√6)(√3) + (-2 √3)(√3) First Inside Outside Last √36 + -2√3 ∙ 6 + √6 ∙ 3 + -2√3 ∙ 3 6 + -2√18 + √18 + -2 ∙ 3 6 – √18 – 6 • √18 = -1 √9 ∙ 2 = -1 ∙ 3 √2 = -3√2

  20. Got it? 3 1. √2(√6 + 5)

  21. Got it? 3 2. (√11 – 2)2

  22. Got it? 3 3. (√6 – 2 √3)(4 √3 + 3 √6)

  23. Problem 4: Cojugates • Examples: • + and - • + 8 and - 8 • What do you notice? • Cojugates: the sum and difference of the same two terms. • ( + )( - ) • 7 – 3 = 4

  24. Problem 4: Rationalizing a Denominator • Multiply by the cojugates. • = • = = 2 + 2

  25. Solving Radical Equations Lesson 10-4

  26. Essential Understanding • Radical equations = equations with a radicand (square root) • Some radical equations can be solved by squaring each side. • The expression under the radicand MUST be positive.

  27. Problem 1 • What is the solution of 7 + = 16? • 7 + = 16 • Subtract 7 from both sides. • = 9 • Square both sides. • ()2= 92 • x = 81

  28. Got it? 1 • What is the solution of -5 + = -2?

  29. Problem 2 • In the expression, t = 2, what is m when t is 3? 3 = 2 1.5 = 1.52 = ()2 2.25 = 2.25 = (2.25)(3.3) = (3.3) 7.425 = m

  30. Problem 3 • What is the solution of ? • ()2 = ()2 • 5t – 11 = t + 5 • 4t – 11 = 5 • 4t = 16 • t = 4

  31. Got it? 3

  32. Extraneous Solutions • Take the original equation x = 3. • Let’s square each side. • x2 = 9 • The solutions would be 3 and -3….right? • However, -3 doesn’t fit in our original equation. • Sometimes when we square each sides, we create a false solution.

  33. Problem 4 • n2 = n + 12 • n2- n – 12 = 0 • (n – 4)(n + 3) = 0 • n = 4 and -3 • Does both numbers work for n?

  34. Problem 4 • (n – 4)(n + 3) = 0 • n = 4 and -3

  35. Problem 5 • What is the solution √3y + 8 = 2? • √3y = -6 • Can you ever have a negative as a product of a square root? • (√3y)2= -62 • 3y = 36 • y = 12 • Check:

  36. Lesson Check 1

  37. Lesson Check 2

  38. Lesson Check 3

  39. Lesson Check 4

  40. Graphing Square Root Functions Lesson 10-5

  41. Square Root Function is… • A function where the “x” or independent variable is under the square root. • The parent square root function is y = .

  42. What is the domain and Range? • Domain: What numbers can you put in for x? • All positive numbers • Range: What kind of numbers are the output of y? • All positive numbers

  43. Problem 1 • What is the domain and range of this function? • y = 2 • Domain: what numbers can you put in for x? • *the radicand can not be negative* • 3x – 9 0 • 3x 9 • x 3 • So, the domain of this function are all real numbers greater or equal to 3. • Range: all positive numbers

  44. Got it? 1 • What is the domain of y = ?

  45. Problem 2 • Graph the function I = , which gives the current I in ampere for a certain circuit with P watts of power. When will the current exceed 2 amperes?

  46. Got it? 2 • When will the current exceed 1.5 amperes?

  47. Problem 3 • Take the parent square root function. y = • If you want to move it UP on the graph, you add OUTSIDE the square root. • If you want to move it DOWN on the graph, you subtract OUTSIDE the square root.

  48. Problem 4 • Take the parent square root function. y = • If you want to move it RIGHT on the graph, you SUBTRACT INSIDE the square root. • If you want to move it LEFT on the graph, you ADD INSIDE the square root.

  49. Got it? 3 and 4 • What coordinate does the graph of y = start on? • What coordinate does the graph of y = start on? • What coordinate does the graph of y = start on? • What coordinate does the graph of y = 4 start on?

  50. Lesson Quiz • Is y = x a square root function? Why or why not? • Can a domain of a square root function include negative numbers? If it can, give an example. If it can not, explain.

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