1. Martin-Gay, Developmental Mathematics 1 AAT-A Date: 12/10/13 SWBAT add and multiply radicals Do Now: Rogawski #77a get the page 224 Complete HW Requests:Adding Subtracting Multiplying Radicals Worksheets Continue Vocab sheet Closure-check answers Students will work pg 254 #43-48 HW: Complete Division of Radicals WS Announcements : Math Team Cancelled Wed. Tutoring: Tues. and Thurs. 3-4 "Do not judge me by my successes, judge me by how many times I fell down and got back up again.“ Nelson Mandela

2. Martin-Gay, Developmental Mathematics 2 Simplifying Radical Expressions

3. Martin-Gay, Developmental Mathematics 3 Rationalizing the denominator -rewrite a radical quotient with the radical confined to ONLY the numerator. There is no radical in the denominator! Process: Multiply the quotient by a form of 1 to eliminate the radical in the denominator. Rationalizing the Denominator

4. Martin-Gay, Developmental Mathematics 4 Rationalize the denominator. Rationalizing the Denominator Example

5. Martin-Gay, Developmental Mathematics 5 To simplify rational quotients with a sum or difference of terms in a denominator, rather than a single radical. Process: Multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing). The conjugate uses the same terms, but the opposite operation (+ or  ). Conjugates

6. Martin-Gay, Developmental Mathematics 6 Rationalize the denominator. Rationalizing the Denominator Example

7. § 15.4 Multiplying and Dividing Radicals

8. Martin-Gay, Developmental Mathematics 8 If and are real numbers, Multiplying and Dividing Radical Expressions

9. Martin-Gay, Developmental Mathematics 9 Simplify the following radical expressions. Multiplying and Dividing Radical Expressions Example

10. § 15.3 Adding and Subtracting Radicals

11. Martin-Gay, Developmental Mathematics 11 Sums and Differences Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences.

12. Martin-Gay, Developmental Mathematics 12 “like” terms- terms with the same variables raised to the same powers can be combined through addition and subtraction. Like radicals are radicals with the same index and the same radicand. Like radicals can be combined with addition or subtraction by using the distributive property. Like Radicals

13. Martin-Gay, Developmental Mathematics 13 Can not simplify Adding and Subtracting Radical Expressions Example

14. Martin-Gay, Developmental Mathematics 14 Simplify the following radical expression. Example Adding and Subtracting Radical Expressions

15. Martin-Gay, Developmental Mathematics 15 Simplify the following radical expression. Example Adding and Subtracting Radical Expressions

16. Martin-Gay, Developmental Mathematics 16 Simplify the following radical expression. Assume that variables represent positive real numbers. Example Adding and Subtracting Radical Expressions

17. Martin-Gay, Developmental Mathematics 17 Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

18. Martin-Gay, Developmental Mathematics 18 The principal (positive) square root is noted as The negative square root is noted as Principal Square Roots

19. Martin-Gay, Developmental Mathematics 19 Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number. Radicands

20. Martin-Gay, Developmental Mathematics 20 Finding the n th root of a number  Finding the square root of a number involves finding a number that, when squared, equals the given number.  In other words, finding such that b 2 = a.  Some vocabulary involved with n th roots: n is the index of the expression. The index tells us what amount of factors we should look for in order to simplify a quantity. Examples: If n = 3, we are looking for some value r such that r 3 = s. If n = 4, we are looking for some value r such that r 4 = s. This is called a radical symbol. s is called the radicand of the radical expression. If the index n is even, then s must be positive. This is because there is no value of r such that r 2 = -s.

21. Martin-Gay, Developmental Mathematics 21 The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number. nth Roots *

22. Martin-Gay, Developmental Mathematics 22 Radicands Example

23. Martin-Gay, Developmental Mathematics 23 Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form. Perfect Squares

24. Martin-Gay, Developmental Mathematics 24 Radicands might also contain variables and powers of variables. To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only. Perfect Square Roots Example

25. Martin-Gay, Developmental Mathematics 25 Simplify the following. nth Roots Example

26. Martin-Gay, Developmental Mathematics 26 The cube root of a real number a Note: a is not restricted to non-negative numbers for cubes. Cube Roots

27. Martin-Gay, Developmental Mathematics 27 Cube Roots Example

28. Martin-Gay, Developmental Mathematics 28 Using the absolute value with radicals Let b = 1, then Now, let b = -1 but To make sure that the answer is positive we add an absolute value. If b is positive there is no problem, however, if b is negative we need |b|

29. § 15.2 Simplifying Radicals

30. Martin-Gay, Developmental Mathematics 30 If and are real numbers, Product Rule and Quotient Rule for Square Roots

31. Martin-Gay, Developmental Mathematics 31 Simplify the following radical expressions. Factor radicand, isolate perfect squares, then simplify No perfect square factor, so the radical is already simplified. Simplifying Radicals Example

32. Martin-Gay, Developmental Mathematics 32 Simplify the following radical expressions. Simplifying Radicals Example

33. Martin-Gay, Developmental Mathematics 33 If and are real numbers, Product and Quotient Rule for Radicals

34. Martin-Gay, Developmental Mathematics 34 Simplify the following radical expressions. Factor radicand, isolate perfect squares, then simplify Simplifying Radicals Example

35. § 15.5 Solving Equations Containing Radicals

36. Martin-Gay, Developmental Mathematics 36 Power Rule (text only talks about squaring, but applies to other powers, as well). If both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution. Extraneous Solutions

37. Martin-Gay, Developmental Mathematics 37 Solve the following radical equation. true Substitute into the original equation. So the solution is x = 24. Solving Radical Equations Example

38. Martin-Gay, Developmental Mathematics 38 Solve the following radical equation. Does NOT check, since the left side of the equation is asking for the principal square root. So the solution is . Substitute into the original equation. Solving Radical Equations Example

39. Martin-Gay, Developmental Mathematics 39 Steps for Solving Radical Equations 1)Isolate one radical on one side of equal sign. 2)Raise each side of the equation to a power equal to the index of the isolated radical, and simplify. (With square roots, the index is 2, so square both sides.) 3)If equation still contains a radical, repeat steps 1 and 2. If not, solve equation. 4)Check proposed solutions in the original equation. Solving Radical Equations

40. Martin-Gay, Developmental Mathematics 40 Solve the following radical equation. true Substitute into the original equation. So the solution is x = 2. Solving Radical Equations Example

41. Martin-Gay, Developmental Mathematics 41 Solve the following radical equation. Solving Radical Equations Example

42. Martin-Gay, Developmental Mathematics 42 Substitute the value for x into the original equation, to check the solution. true false So the solution is x = 3. Example continued Solving Radical Equations

43. Martin-Gay, Developmental Mathematics 43 Solve the following radical equation. Solving Radical Equations Example

44. Martin-Gay, Developmental Mathematics 44 Substitute the value for x into the original equation, to check the solution. false So the solution is . Example continued Solving Radical Equations

45. Martin-Gay, Developmental Mathematics 45 Solve the following radical equation. Solving Radical Equations Example

46. Martin-Gay, Developmental Mathematics 46 Substitute the value for x into the original equation, to check the solution. true So the solution is x = 4 or 20. Example continued Solving Radical Equations

47. § 15.6 Radical Equations and Problem Solving

48. Martin-Gay, Developmental Mathematics 48 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. (leg a) 2 + (leg b) 2 = (hypotenuse) 2 The Pythagorean Theorem

49. Martin-Gay, Developmental Mathematics 49 Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches. c 2 = 2 2 + 7 2 = 4 + 49 = 53 c = inches Using the Pythagorean Theorem Example

50. Martin-Gay, Developmental Mathematics 50 By using the Pythagorean Theorem, we can derive a formula for finding the distance between two points with coordinates (x 1,y 1 ) and (x 2,y 2 ). The Distance Formula

51. Martin-Gay, Developmental Mathematics 51 Find the distance between (  5, 8) and (  2, 2). The Distance Formula Example

52. Martin-Gay, Developmental Mathematics 52 15.1 – Introduction to Radicals 15.2 – Simplifying Radicals 15.3 – Adding and Subtracting Radicals 15.4 – Multiplying and Dividing Radicals 15.5 – Solving Equations Containing Radicals 15.6 – Radical Equations and Problem Solving Chapter Sections