1 rules of radicals x

rules of radicals

1. Rules of Radicals

2. Rules of Radicals For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0.

3. Square Rule:  x 2 =  x  x = x Rules of Radicals For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0.

4. Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0.

5. Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y We use these rules to “simplify” root-expressions. Rules of Radicals For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. Example A. Simplify  8 = b.  72 = d.  x 2 y 3 = c.  x 2 y =

6. Example A. Simplify  8 = Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 = d.  x 2 y 3 = c.  x 2 y = For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

7. Example A. Simplify  8 =  4  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 = d.  x 2 y 3 = c.  x 2 y = For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

8. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 = d.  x 2 y 3 = c.  x 2 y = For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

9. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 =  36  2 d.  x 2 y 3 = c.  x 2 y = For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

10. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 = c.  x 2 y = For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 4 , y 2 , .. etc , (and only the square factors) to extract .

11. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 = c.  x 2 y =  x 2  y For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 extract the square factor x 2 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

12. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 = c.  x 2 y =  x 2  y = x  y For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 extract the square factor x 2 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

13. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 =  x 2 y 2  y c.  x 2 y =  x 2  y = x  y For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 extract the square factor x 2 extract the square factor x 2 y 2 We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract .

14. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract . Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 =  x 2 y 2  y = xy  y c.  x 2 y =  x 2  y = x  y For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 extract the square factor x 2 extract the square factor x 2 y 2

15. Example A. Simplify  8 =  4  2 = 2  2 Square Rule:  x 2 =  x  x = x Multiplication Rule:  x · y =  x ·  y We use these rules to “simplify” root-expressions. Specifically, to simplify a square-root, look for square factors 4, 9, 16, 25, 36 , … x 2 , y 4 , .. etc , (and only the square factors) to extract . Rules of Radicals b.  72 =  36  2 = 6  2 d.  x 2 y 3 =  x 2 y 2  y = xy  y c.  x 2 y =  x 2  y = x  y A radical expression is said to be simplified if as much as possible is extracted out of the square-root. For the following discussion of square roots, variables such as x , y, z, .. are assumed ≥ 0. extract the square factor 4 extract the square factor 36 extract the square factor x 2 extract the square factor x 2 y 2

16. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Rules of Radicals

17. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 Rules of Radicals

18. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 Rules of Radicals

19. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) Rules of Radicals

20. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 Rules of Radicals

21. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 Rules of Radicals

22. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) Rules of Radicals

23. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 Rules of Radicals

24. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y Rules of Radicals

25. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals

26. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  =

27. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify.

28. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  a.

29. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  9  4  a. =

30. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  9  4 3 2  a. = =

31. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  9  4 3 2 9y 2 x 2  a. = =  b.

32. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  9  4 3 2 9y 2 x 2  9y 2  x 2  a. = =  b. =

33. A radical expression is said to be simplified if as much as possible is extracted out of the square-root. Example B. Simplify. a.  72 =  4  18 = 2  18 (not simplified yet) = 2  9  2 = 2 * 3 *  2 = 6  2 (simplified) b.  80x 4 y 5 =  16 ·5 x 4 y 4 y = 4x 2 y 2  5y Rules of Radicals Division Rule: y x  y  x  = Example C. Simplify. 9 4  9  4 3 2 9y 2 x 2  9y 2  x 2 3y x  a. = =  b. = =

34. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, Rules of Radicals

35. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. Rules of Radicals

36. The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals

37. Example D. Simplify 5 3 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. 

38. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =  

39. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =   = 25 15  

40. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =   = 25 15   = 5 15 

41. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =   = 25 15   = 5 15  8x 5 b.  5 1  15 or

42. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   5 1  15 or

43. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   5 1  15 or

44. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   5 1  15 or

45. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * 5 1  15 or

46. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  5 1  15 or

47. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  5 1  15 or 4x 1  10x or

48. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  WARNING!!!!  a ± b =  a ±  b 5 1  15 or 4x 1  10x or

49. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  WARNING!!!!  a ± b =  a ±  b For example:  4 + 9  13 = 5 1  15 or 4x 1  10x or

50. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  WARNING!!!!  a ± b =  a ±  b For example:  4 + 9  13 = 5 1  15 or 4x 1  10x or

51. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  WARNING!!!!  a ± b =  a ±  b For example:  4 + 9 =  4 +  9  13 = 5 1  15 or 4x 1  10x or

52. Example D. Simplify 5 3 5 · 5 3 · 5 The radical of a fractional expression is said to be simplified if the denominator is completely extracted out of the radical, i.e. the denominator is radical free. If the denominator does contain radical terms, multiply the top and bottom by suitably chosen quantities to remove the radical term in the denominator to simplify it. Rules of Radicals 2 a. =   = 25 15   = 5 15  8x 5 4 ·2 x 5 b. =   = 2x 5   = 2 2x 5   2x 2x   = 2 2x 10x  * = 4x 10x  WARNING!!!!  a ± b =  a ±  b For example:  4 + 9 =  4 +  9 = 2 + 3 = 5  13 = 5 1  15 or 4x 1  10x or

53. Rules of Radicals Exercise A . Simplify the following radicals. 1 .  12 2 .  18 3 .  20 4 .  28 5 .  32 6 .  36 7 .  40 8 .  45 9 .  54 10 .  60 11 .  72 12 .  84 13 . 90 14 .  96x 2 15 .  108x 3 16 .  120x 2 y 2 17 .  150y 4 18 .  189x 3 y 2 19 .  240x 5 y 8 18 .  242x 19 y 34 19 .  12  12 20 .  18  18 21 . 2 16 23 . 183 22 . 123 24 . 1227 25 . 1850 26 . 1040 27 . 20x15x 28 . 12xy15y 29 . 32xy 3 24x 5 30 . x 8 y 13 x 15 y 9 Exercise B . Simplify the following radicals. Remember that you have a choice to simplify each of the radicals first then multiply, or multiply the radicals first then simplify.

54. Rules of Radicals Exercise C . Simplify the following radicals. Remember that you have a choice to simplify each of the radicals first then multiply, or multiply the radicals first then simplify. Make sure the denominators are radical–free. 8x 5 31.  x 10  14 5x 32.  7 20  5 12 33.  15  8x 5 34.  3 2  3 32x 35.  7 5  5 2 36.  29 x  x (x + 1) 39.  x (x + 1)  x (x + 1) 40.  x(x + 1) 1  1 (x + 1) 37.  x (x 2 – 1) 41.  x(x + 1) (x – 1)  x (x + 1) 38.  x 2 1 –  1 Exercise D. Take the denominators of out of the radical. 42. 9x 2 1 –  1 43.