Radical Expressions

1. Radical Expressions • MA.912.A.6.1 Simplify radical expressions. • MA.912.A.6.2 Add, subtract, multiply, and divide radical expressions

2. What is a radical? Radical Vinculum • This is the symbol for square root. The implied index is 2, however it does not need to be written. 2 Root Index • If the index is 3, it is called a 3rd root, or cube root. Radicand

3. nth roots *Notice how the index is invisible in the square root.

4. What is a square root? The number c is a square root of the number a if c2 = a. The definition is saying that the number 6 is a square root of 36, since 62 = 36. But since (− 6)2 = (− 6)(− 6) = 36 , it follows by the above definition that − 6 is also a square root of 36. Thus, 36 has two square roots: a positive one and a negative one.

5. What is a square root? The notation used to indicate the square root of some number “a” is . When a square root is written with the radical symbol we are referring to the positive, or principal square root. For example, = 6 . We will adopt the notation of to represent the negative square root of a number “a”. For example, = − 6 . When using words… “What is the square root of…”, then the answer should indicate both roots.

6. Perfect Squares 12=1, 22= 4, 32= 9, 42= 16, 52= 25, 62= 36, 72= 49, 82= 64, 92= 81, 102= 100, 112= 121, 122= 144, 132= 169, 142= 196, 152= 225… 202=400

7. Evaluate 8 Area = 64 sq units 8

8. Approximating a Square Root Approximate the square root to the nearest integer 122= 144 132= 169 Since 149 is between 144 and 169, the square root of 149 is Between 12 and 13.

9. Approximating a Square Root Approximate the square root to the nearest integer 72= 49 82= 64 Since 57 is between 49 and 64, the square root of 57 is Between 7 and 8.

10. Estimate the square root of 33 • Add on another row and column so that you now have • 6 x 6 box to represent the • square root of 36. You know that it is going to be between 5 and 6 since 33 falls between 25 and 36. Draw a 5 x 5 box to represent the square root of 25. Shade in the 25 boxes.

11. Estimate the square root of 33 • 8 out of the 11 extra boxes are shaded, this represents 8/11 • The square root of 33 is • estimated to Shade in the extra boxes so you have 33 colored in.

12. Estimate the square root of 85 You know that it is going to be between 9 and 10 since 85 falls between 81 and 100. Draw a 9 x 9 box to represent the square root of 81. Shade in the 81 boxes. Add on another row and column so that you now have 10 x 10 box to represent the square root of 100.

13. Estimate the square root of 85 • 4 out of the 19 extra boxes are • shaded, this represents 4/19 • The square root of 85 is • estimated to Shade in the extra boxes so you have 85 colored in.

14. Simplify Click link below to graph or see next slide. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

17. Simplify Are absolute value symbols necessary? NO!When evaluating an even rootand the resulting answer has an even exponent, absolute value symbols are not necessary.

18. For any numbers a and b where and , Product Property of Radicals

19. Simplify The following statement is found in most textbooks: “In this book, all variables in radical expressions represent non-negative numbers” This statement eliminates the need to use the absolute value symbol.

20. Simplifying Square Roots We can use the Product Property of Radicals to simplify radical expressions. Look for a perfect square factor A square root is simplified if there are no perfect square factors in the radicand.

21. Simplifying Square Roots

22. Simplifying Square Roots

23. Simplifying Square Roots You can use divisibility rules and prime factoring to help with simplifying square roots. 2 363 121 3 11 11

24. Multiplying Radicals or

25. Multiplying Radicals

26. Multiplying Radicals

27. Multiplying Radicals

28. For any numbers a and b where and , Quotient Property of Radicals

29. Dividing Radicals

30. Dividing Radicals

31. Simple Radical FormA radical is said to be in simple radical form if: • There are no radicals in the denominator • The radicand(s) in the numerator are as small as possible. Not simplified radical form:

32. The Process of eliminating a radical from the denominator is called: Rationalizing the Denominator • To rationalize the denominator, you must convert the radicand into a perfect square number. In this case you can do it by multiplying by ____ . • In order not to change the value of the original number, you must multiply the numerator by_____ as well.

33. Rationalizing the Denominator Why?

34. Express in simple radical form reduce Now Rationalize! reduce Sometimes it is more efficient to simplify the radical before you rationalize.

35. Express in simple radical form Sometimes it is more efficient to reduce the fraction before you rationalize.

36. Like Radicals Two radicals are called “like radicals” if they have the same index and radicand. Are the examples below like radicals? NO, the indexes are not the same. YES, the radicals have the same index and radicand.

37. Adding & Subtracting Radicals • When adding or subtracting radicals, you will use the same concept as that of adding or subtracting "like" variables. • In other words, the radicals must be “like radicals” before you add (or subtract) them.

38. Adding & Subtracting Radicals • Procedure: • Simplify all given radicals. • Are the radicals like radicals? • Leave the expression as a sum or difference. • Use the distributive • property to add/subtract. • Yes • No

39. Adding & Subtracting Radicals ? ? ? ? ?

41. Simplify the radical expression or

42. Simplify the radical expression

43. Simplify the radical expression

44. Simplify the radical expression

45. EOC Practice

46. EOC Practice

47. Game Time! http://www.math-play.com/square-roots-game.html