1. TRIGONOMETRY
5.3 RADICALS
Ms. Albarico

2. Radicals Students are expected to:
* Demonstrate an understanding of the role of irrational numbers in applications.
* Approximate square roots.
* Demonstrate an understanding of and apply properties to operations involving square roots.
* Apply the Pythagorean Theorem.
* Use inductive and deductive reasoning when observing patterns, developing properties, and making conjectures.

3. Perfect Squares 1
225 64 4 81
256
100
289 9
121 16
324
144 25
361
169 36
400
196 49
625

4. Simplify
= 2
= 4
This is a piece of cake!
= 5
= 10
= 12

5. What are radicals?
A Radical is the root of a number.

6. What are radicals?
A Radical is the root of a number.

7. What are radicals? A Radical is the root of a number. coefficient

8. What are radicals? A Radical is the root of a number. For example:
radical sign
coefficient
radical

9. What are radicals? A Radical is the root of a number. For example:
coefficient
radical

10. What are radicals? A Radical is the root of a number. For example:
5 is the square root of 25 because 5x5 = 25
coefficient
radical

11. What are radicals? A Radical is the root of a number. For example:
5 is the square root of 25 because 5x5 =25
coefficient
radical

12. What are radicals? A Radical is the root of a number. For example:
5 is the square root of 25 because 5x5 = 25
2 is the cubed root of 8 because 2x2x2 = 8.
coefficient
radical

14. Determining Roots
You can determine the root using mental math.

15. Determining Roots You can determine the root using mental math.
Example:

16. Determining Roots You can determine the root using mental math.
Example:

17. Determining Roots You can determine the root using mental math.
Example:
You can also find the root using your calculator.

18. Determining Roots You can determine the root using mental math.
Example:
You can also find the root using your calculator.
Example:

19. Determining Roots You can determine the root using mental math.
Example:
You can also find the root using your calculator.
Example:

20. Determining Roots You can determine the root using mental math.
Example:
You can also find the root using your calculator.
Example:
Be careful not to hit the wrong button on your calculator!

21. Let’s investigate! 1)

22. 2)

23. 3)
I. Solve for the hypotenuse of the selected triangle in your geoboard paper.
II. At the back of your paper, draw a square with an area of: 34 41 52 29

24. Simplifying
Square Roots

25. Simplify = = = = = = = = = = Perfect Square Factor * Other Factor
LEAVE IN RADICAL FORM = = = =

26. Simplify = = = = = = = = = = Perfect Square Factor * Other Factor
LEAVE IN RADICAL FORM = = = =

27. Simplify = = = = = = = = = = Perfect Square Factor * Other Factor
LEAVE IN RADICAL FORM = =

28. Adding and Subtracting Radicals
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.

29. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.

30. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.

31. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.
Example 2:

32. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.
Example 2:

33. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.
Example 2:
Example 3:

34. Adding and Subtracting Radicals
Example 1:
Only like radicals can be added or subtracted.
Like radicals can have different coefficients as long as the radicals are the same.
Subtract the coefficients but keep the same radical.
You may have to simplify your question first, to get a like radical.
Example 2:
Example 3:

35. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.

36. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.

37. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.
Always make sure that the radicals are similar.

38. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.
Always make sure that the radicals are similar.
Another common mistake is forgetting to simplify.

39. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.
Always make sure that the radicals are similar.
Another common mistake is forgetting to simplify.

40. Common Mistakes Made While Adding and Subtracting Radicals
One common mistake is adding without having like radicals.
Always make sure that the radicals are similar.
Another common mistake is forgetting to simplify.
This question is not done! The answer is:

41. + To combine radicals: combine the coefficients of like radicals
Combining Radicals +
To combine radicals: combine the coefficients of like radicals

42. Simplify each expression:

43. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.

44. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.

45. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.
Example:

46. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.
Example:

47. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.
Example:

48. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.
Example:

49. Common Mistakes made while doing combined operations
Be careful to multiply correctly.

50. Common Mistakes made while doing combined operations
Be careful to multiply correctly.
Correct Answer:

51. Common Mistakes made while doing combined operations
Be careful to multiply correctly.
Correct Answer:

52. Common Mistakes made while doing combined operations
Be careful to multiply correctly.
Correct Answer:
Incorrect Answer:

53. Common Mistakes made while doing combined operations
Be careful to multiply correctly.
Correct Answer:
Incorrect Answer:

54. Simplify each expression: Simplify each radical first and then combine.

55. Simplify each expression: Simplify each radical first and then combine.

56. Simplify each expression.
Lets Do Some Work!
Simplify each expression.

63. Multiplying Radicals
Steps
Example:

64. Multiplying Radicals
Steps
Example:

65. Multiplying Radicals
Steps
Example:
Multiply coefficients together 1.

66. Multiplying Radicals Steps Example: 1. 2.
Multiply coefficients together
Multiply radicals together 1. 2.

67. Multiplying Radicals Steps Example: Multiply coefficients together
Multiply radicals together
Simplify radicals if possible

68. Examples of Multiplying Radicals
Correct example:

69. Examples of Multiplying Radicals
Correct example:

70. Examples of Multiplying Radicals
Correct example:

71. Examples of Multiplying Radicals
Correct example:
Be careful to multiply the coefficients with coefficients and radicals with radicals.

72. Examples of Multiplying Radicals
Correct example:
Be careful to multiply the coefficients with coefficients and radicals with radicals.
Incorrect Example:

73. Examples of Multiplying Radicals
Correct example:
Be careful to multiply the coefficients with coefficients and radicals with radicals.
Incorrect Example:

74. Examples of Multiplying Radicals
Correct example:
Be careful to multiply the coefficients with coefficients and radicals with radicals.
Incorrect Example:
This is Wrong!
Don’t multiply these together!

75. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.

76. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).

77. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).

78. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).
Rationalize denominators

79. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).
Rationalize denominators.
Reduce coefficients again.

80. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).
3. Rationalize denominators.
4. Reduce coefficients again.
Example:

81. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).
Rationalize denominators.
Reduce coefficients again.
Example:

82. Dividing Radicals (Part 1)
There are 4 steps to dividing radicals.
1. Reduce coefficients and/or radicals by common factor (if you can).
2.Take out any perfect squares (if possible).
Rationalize denominators.
Reduce coefficients again.
Example:
Divide top and bottom by

83. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.

84. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.
Example of this error:

85. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.
Example of this error:

86. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.
Example of this error:
The correct answer to this question is:

87. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.
Example of this error:
The correct answer to this question is:

88. Common Errors Done While Dividing Radicals
Be careful not to divide a coefficient by a radical.
Example of this error:
The correct answer to this question is:
Multiply top and
bottom by

89. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply both sides by the conjugate.

90. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.

91. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Binomial denominator.

92. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Example:
Binomial denominator.

93. Dividing Radicals (part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Example:
Binomial denominator.

94. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Example:
Multiply numerator and denominator by
Binomial denominator.

95. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Example:
Multiply top and bottom by
Binomial denominator.

96. Dividing Radicals (Part 2)
When rationalizing binomial denominators, multiply top and bottom by the conjugate.
Example:
Multiply top and bottom
Binomial denominator.

97. Homework
In your notebook, answer CYU # 7, 8, 9, 13, and 14 on pages

98. Extra Practice for Quiz!
Solve without a calculator. 4. 1.
Simplify. 2. 5.
Use your calculator to solve. 6. 3.

99. Extra Practice for Quiz!
Solve.
Multiply.
11. 7.
Divide.
12. 8.
Add or Subtract.
13. 9.
10.

100. Answers!
1. 5
2. 1.1
3. 492
4. 1.5 5. 6.
7. 24 8. 9.
10.

101. Answers!
11.
12.
13.