5.02   Cube Roots and More

           from Basic Algebra: One Step at a Time © 2002

P. 403-412

Dr. Robert J. Rapalje

Seminole State College of Florida

 

ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE

 

To see selected solutions in Living C O L O R  click here!

 

In the previous section, you learned that the expression or  means the square root of x.  This essentially means, “What squared would equal x?”  The quantity inside the radical sign (in this case x) is called the radicand, and the 2 (in this case) is called the index of the radical.  Now, the expression  is called the cube root of x, and it asks the question, "What cubed would equal x?"  Likewise, means the fourth root of x, means the fifth root of x, etc.  In general,  means the nth root of X, where the radicand is X, and the index of the radical is n. 

The operations of square root, cube root, fourth root, etc. are actually inverse operations for the operations of squaring, cubing, raising to the fourth power, etc.  When taking square roots in the last section, it was essential to be familiar with the perfect squares:  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and 169.  Also, remember that the even powers (x2, x4, x6, x8, x10, etc.) were and are perfect squares.  Now, when taking a cube root, it is essential to be familiar with (i.e., memorize them!!) the perfect cubes, and other powers, especially the numbers  1, 8, 27, 64, and 125.   

 23  =        8                       24  =       16                       25  =       32

 33  =      27                       34  =       81

 43  =      64                 

 53  =    125

And again, the list goes on.  However, these are the main numbers that we use, and with which you need to be familiar.  You really need to have the numbers 8, 27, 64, and 125 in your head before you continue this lesson!!

 

Taking a cube root of a number is actually the inverse operation of cubing.   Suppose you cubed the number 5.  The answer of course is 125.  Now, what would you have to do to the 125 to get back to the 5?  You would take the cube root of 125, written, which is 5.

NOTE:            Frequently the terminology “square root” and “radical” are used interchangeably.  They are NOT the same.  The term “radical” may be used generally to refer to a square root, cube root, etc.  The term “square root” does not include cube roots, fourth roots, etc.

EXAMPLE 1.             Fill in the blanks below to find .

= ______ because  (        )3 = _______.   

Solution:                     =      2     because   (   2   )3 =      8     .

Therefore, the cube root of 8 or  is 2.

 

EXAMPLE 2.             Fill in the blanks below to find .

= ______ because  (        )4 = _______.   

Solution:                     =      3     because   (   3   )4 =     81     .

Therefore, the fourth root of 81 or  is  3.

 

EXERCISES. Fill in the blanks to find the radical expressions.

 1.    =  _______  because  (          )3  =  27.

 2.    =  _______  because  (          )3  =  64.   

 3.      =  _______  because  (          )3  =  ______.  

 4.      =  _______  because  (          )3  =  ______.  

 5.   =  _______  because  (          )3  =  ______.  

 6.       =  _______  because  (          )3  =  ______.  

 7.     =  _______  because  (          )4  =  ______.  

 8.     =  _______  because  (          )4  =  ______.  

 9.     =  _______  because  (          )5  =  ______.  

10.      =  _______  because  (          )5  =  ______.  

 

The following exercises are repeated for extra practice!

11.                          12.                        13.                    14.            

 

15.                          16.                           17.                    18.                      

 

In the next exercises, you will need to remember and use the law of exponents: (xm)n = xmn .  When you raise an exponent  to a power, you multiply the exponents.  In particular, when you cube an exponent, you multiply the exponent times 3, when you raise a power to the fourth power, you multiply the exponent times 4, when you raise to the fifth power, you multiply times 5, etc..

 

EXAMPLE 3.      Simplify     (x4)3                         EXAMPLE 4.    Simplify       (x5)4  

Solution:                               (x4)3 = x12                Solution:                                 (x5)4 = x20

 

EXAMPLE 5.     Complete the blanks in order to find.

     = _______   because (          )3 = ________.

Solution:              =       x4          because (  x4  )3   =       x12     .

                             Therefore, the cube root of x12  (or )  is    x4  .

 

EXAMPLE 6.     Complete the blanks in order to find.

    = _______   because (          )4 = ________.

Solution:             =       x5          because (  x5  )4   =       x20     .

                            Therefore, the fourth root of x20  (or )  is    x5  .

 

EXERCISES. Complete the blanks in order to simplify the radicals.

19.  = _____       because (        )3 = ________. 

 

20.  = _____       because (        )3 = ________.

 

21.  = _____       because (        )4= ________. 

 

22.  = _____       because (        )5 = ________.  

 

Simplify each of the following:

23.             24.               25.               26.              27.              28.  

 

29.            30.               31.                 32.             33.              34.  

 

35.                     36.                             37.                       38.

 

 

39.                      40.                            41.                      42.

 

 

The Product Property of Square Roots from the last section can be extended to cube roots, fourth roots, and in general nth roots.  When extended beyond square roots, we call this the Product Property of Radicals.  This property can often be used to simplify radicals involving cube roots, fourth roots, etc. 

 

 

As with the product property of square roots,  the product property of radicals is a property of real numbers.  Therefore, if the index of the radical is even, then the radicands must be positive.  The same “radical two step” method still applies, except that the second step is now “cube root,” “fourth root,” etc.  In the first step, you must "sort" the radical, placing the "perfect powers" in the first radical and the other "leftover" factors in the second radicals.  In the second step, you take the appropriate root of the perfect power, and just bring down the "leftover" radical. 

EXAMPLE 7.    Simplify    and calculate its value to the nearest hundredth.

Step 1.            Find a perfect cube that is a factor of 24, and write 24 as a product.  The perfect cube is 8; the “left-over” factor is 3, and the product is 8×3.

   

Step 2.              Take the cube root of 8, which is 2.  Keep the cube root sign on the 3.

 .

Use your calculator (see Section 1.04),  is approximately 2.88449914, which rounds to  2.88.

 

EXAMPLE 8.    Simplifyand calculate its value to the nearest hundredth.

Step 1.            Find a perfect cube that is a factor of 56, and write as a product.  The perfect cube is 8, the leftover factor is 7, and the product is 8×7.

                       

Step 2.             Take the cube root of 8, which is 2.  Keep the cube root sign on the 7.

                         or

The decimal approximation is 3.82586236554, which rounds to 3.83.

 

EXAMPLE 9.    Simplify and calculate its value to the nearest hundredth.

Step 1.            Find a perfect cube that is a factor of 250, and write as a product.  The perfect cube is 125, the leftover factor is 2, and the product is 125×2.

                         .

Step 2.             Take the cube root of 125, which is 5.  Keep the cube root sign on the 5.

                         , which is approximately 6.30.

 

EXERCISES.     Simplify the radicals completely.  Find the values to the nearest hundredth.

 43.                                            44.                                  45.    

                                                                    

 ____________                                 ____________                         _____________

 

46.                                          47.                                   48.      

                                                                      

 ____________                                ____________                         _____________

 

49.                                            50.                                    51.      

                                                                  

 ____________                                ____________                         _____________

 

52.                                           53.                                 54.      

____________                                ____________                      _____________

 

____________                                ____________                      _____________

 

55.                                             56.                                    57.      

     ____________                           ____________                       _____________

 

     ____________                           ____________                       _____________

 

58.                                              59.                                 60.       

     ____________                           ____________                       _____________

 

     ____________                           ____________                       _____________

 

 

61.                                                                           62.                                 

     ____________                                                      ____________                   

 

     ____________                                                    ____________                 

 

63.                                                                          64.                      

     ____________                                                      ____________

 

     ____________                                                      ____________                   

 

EXTRA CHALLENGE

EXAMPLE 10.           Simplify and calculate its value to the nearest hundredth.

 

Step 1.            Find a perfect fourth power that is a factor of 320, and write as a product.  The perfect fourth power is 16, the leftover factor is 20, and the product is 16×20.

.

Step 2.            Take the fourth root of 16, which is 2.  Keep the fourth root sign on the 20.

, which is approximately 4.23.

 

EXERCISES.             Simplify the following radicals. 

65.                                                  66.                                  67.      

                                                                       

    ____________                              ____________                    _____________

 

68.                                               69.                                    70.      

                                                                     

     ____________                              ____________                    _____________

 

71.                                                  72.                                  73.      

 

 

 

 

74.                                                75.                                  76.      

 

 

 

EXAMPLE 11.           Simplify.

Step 1.            Find all perfect cube factors of 108x6y14, and write as a product.  The perfect cube is 27, with a leftover factor of 4. Also, x6 and y12 are perfect cubes, leaving y2 as leftover.  The product is 27x6y12 × 4y2 .

.

Step 2.             Take the cube root of the first part, and keep the cube root sign on the second.

 .

EXERCISES.          Simplify the radicals completely.  Find the values to the nearest hundredth.

77.                                                      78.                                      

                                                                  

   _______________                                 ________________                  

 

79.                                                    80.  

 

  

81.                                                     82.        

 

 

 83.                                                     84.                                    

 

 

85.                                                        86.      

                                                            

_______________                                      ________________                  

 

87.                                                     88.    

 

 

89.                                                   90.  

 

 

91.                                                     92.  

 

 

93.                                                  94.  

 

 

 

ANSWERS 5.02

p. 404 - 412:

 1. 3; 2. 4; 3. 1; 4. 0; 5. 5; 6. 2; 7. 2; 8. 3;  9. 2; 10. 1; 11. 4; 12. 5; 13. 2; 14. 0; 15. 3; 16. 2; 17. 2; 18. 3; 19. x5; 20. x10; 21. x6; 22. x3; 23. x2; 24. x8; 25. x6; 26. x10; 27. x8; 28. x6; 29. x4; 30. x5; 31. x2; 32. x4; 33. x5; 34. x3; 35. 5x2; 36. 2x4; 37. 3x9;     38. 4x17; 39. 2x4; 40. 3x3; 41. 2x4; 42. 2x12; 43.; 44.; 45.; 46.; 47.; 48.; 49.;  50.; 51.; 52.; 53.; 54.; 55.; 56.; 57.; 58.; 59.; 60.; 61.; 62.; 63.; 64.; 65.; 66.; 67.; 68.; 69.; 70.; 71.; 72.; 73.; 74.; 75.;  76.;  77.;  78.;  79.; 80.; 81.; 82.;  83.; 84.; 85.; 86.; 87. ; 88.; 89.; 90.; 91.; 92.; 93.; 94..

 

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