3.02  Operations with Radicals

Intermediate Algebra: One Step at a Time.  Page 250- 255:   #16, 17, 19, 20, 21, 23, 24, 38, 41, 42, 44, 45, 48, 51, 52, 55, 58, 60, 64

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

Simplifying radicals is not nearly as hard as you think it is, especially if it is explained in living color!  Consider these exercises.  Notice how the colors make the exercises easier to follow.  Can you imagine what this would look like in black and white?  Most of our television is in color--why not math? 

First, before you even start to do #16, there are cube roots, so you must get the perfect cubes in your mind:        

2³=8,    3³=27,    4³=64,    5³=125. 

16.      5  −  4

Find a perfect cube that divides into 108 (that would be 27) and a perfect cube that divides into 32 (that would be 8).  108=27x4 and 32=8x4.

            5  −  4

            5  −  4

            5 •  −  4  •

            5 • 3  •  −  4 • 2  •

Now multiply the numbers 5 times 3 and the 4 times 2.

            15     −      8

Now you have like terms so you can combine the 15 and the -8, and the final answer is  7  

In # 17, you have 4th roots, so keep in mind that 2⁴=16 and 3⁴=81.

17.      7  − 3

Find a perfect 4th power that divides into 32 (that would be 16) and a perfect 4th power that divides into 162 (that would be 81).  32=16x2 and 162=81x2.

            7  − 3

            7  − 3

             7 •  − 3 •

             7 •  2 •  − 3 • 3  •

Now multiply the numbers 7 times 2 and the 3 times 3.

            14  –  9

Now you have like terms so you can combine the 14 and the -9, and the final answer is

19.     

First, separate each of the square roots into two square roots.  Sort out the square roots into perfect squares that go in the first (red) square root, and the left-over factors that go in the second (blue) square root.

Everyone can take the square root of the first (red) radicals since they are perfect squares.  Nobody knows what to do about the second (blue) radical since they cannot be simplified.  So do what you can do (the red radicals), and leave the rest (blue radicals!) alone:

As in the last step, you do what you are able to do next--multiply outside the radicals:

Notice that what is left turn out to be like radicals and like terms.  They combine together:

20.  

First, separate each of the square roots into two square roots.  Sort out the square roots into perfect squares that go in the first (red) square root, and the left-over factors that go in the second (blue) square root.

Everyone can take the square root of the first (red) radicals since they are perfect squares.  Nobody knows what to do about the second (blue) radical since they cannot be simplified.  So do what you can do (the red radicals), and leave the rest (blue radicals!) alone:

As in the last step, you do what you are able to do next--multiply outside the radicals:

Notice that what is left turn out to be like radicals and like terms.  They combine together:

21.  

First, separate each of the square roots into two square roots.  Sort out the square roots into perfect squares that go in the first (red) square root, and the left-over factors that go in the second (blue) square root.

Everyone can take the square root of the first (red) radicals since they are perfect squares.  Nobody knows what to do about the second (blue) radical since they cannot be simplified.  So do what you can do (the red radicals), and leave the rest (blue radicals!) alone:

As in the last step, you do what you are able to do next--multiply outside the radicals:

Notice that what is left turn out to be like radicals and like terms.  In fact, they subtract out, and the final answer is

23.

First, separate each of the cube roots into two cube roots.  Sort out each cube roots into perfect cubes that go in the first (red) cube root, and the left-over factors that go in the second (blue) cube root.

Everyone can take the cube root of the first (red) radicals since they are perfect cubes.  Nobody knows what to do about the second (blue) radical since they cannot be simplified.  So do what you can do (the red radicals), and leave the rest (blue radicals!) alone:

Do what you are able to do next--multiply outside the radicals:

Notice that what is left turn out to be like radicals and like terms.  Combining like terms, you have

      Final Answer!!

24.   

First, separate each of the cube roots into two cube roots.  Sort out each cube roots into perfect cubes that go in the first (red) cube root, and the left-over factors that go in the second (blue) cube root.

Everyone can take the cube root of the first (red) radicals since they are perfect cubes.  Nobody knows what to do about the second (blue) radical since they cannot be simplified.  So do what you can do (the red radicals), and leave the rest (blue radicals!) alone:

As in the last step, you do what you are able to do next--multiply outside the radicals:

Notice that what is left turn out to be like radicals and like terms.  Combining like terms, you have

            Final Answer!!

38.     

First, you can multiply the numbers together and place the result within a single square root.  However, if you do this, you will get one very large number that will be difficult to break down.  So, instead of multiplying , write the numbers in factored form:

Next, notice that something “special” happened in this last step!  You have a “pair” of  factors.  Now, you can rewrite this as a product of two radicals in which you place the perfect squares in the first (red) radical, and the “left over” factor(s) in the second (blue) radical.  The trick here is to recognize that the pair of  factors is a square, but also the  is a perfect square, and these must all go in the first (red) radical.

        Final Answer!!

Since this is a numerical problem, it can be checked with a calculator.

Calculate the value of the problem:  =  138.6363146

Calculate the value of the answer:               =  138.6363146.   It checks!

41.     

First, you can multiply the numbers together and place the result within a single cube root.  However, if you do this, you will get one very large number that will be difficult to break down.  So, instead of multiplying , write the numbers in factored form:

Next, sort out these factors, placing any perfect cubes (that is, three of a kind!) in the first (red) cube root, and place the left-over factors in the second (blue) cube root.   Notice that you have three factors of 5, so put these in the first cube root.  What is "left-over" is a pair of 3's, so they go in the second cube root.

     Final Answer!!

Since this is a numerical problem, it can be checked with a calculator.

Calculate the value of the problem:    =  10.40041912

Calculate the value of the answer:            =  10.40041912.  It checks!!

42.      

First, you can multiply the numbers together and place the result within a single cube root.  However, if you do this, you will get one very large number that will be difficult to break down.  So, instead of multiplying , write the numbers in factored form:

Next, sort out these factors, placing any perfect cubes (that is, three of a kind!) in the first (red) cube root, and place the left-over factors in the second (blue) cube root.  Notice that you have three factors of 3, so put these in the first cube root.  What is "left-over" is a pair of 5's and a factor of 7, so they go in the second cube root.

     Final Answer!!

Since this is a numerical problem, it can be checked with a calculator.

Calculate the value of the problem:  =  16.78033413

Calculate the value of the answer:         =  16.78033413.  It checks!

44.     

First, you can multiply the numbers together and place the result within a single cube root.  However, if you do this, you will get one very large number that will be difficult to break down.  So, instead of multiplying , write the numbers in factored form:

Next, sort out these factors, placing any perfect cubes (that is, three of a kind!) in the first (red) cube root, and place the left-over factors in the second (blue) cube root.   Notice that you have three factors of 11, so put these in the first cube root.  What is "left-over" are factors of 2 and 5, so they go in the second cube root.

     Final Answer!!

Since this is a numerical problem, it can be checked with a calculator.

Calculate the value of the problem:     =  23.69878159

Calculate the value of the answer:            =  23.69878159.   It checks!

45.

Remember, you multiply the numbers that are OUTSIDE the radical together, and you keep them OUTSIDE the radical.  Then you multiply the numbers that are INSIDE the radical together and keep them INSIDE the radical.

Now, simplify the radical 45.  Break it down into 9 times 5.

        FINAL ANSWER!

Since this is a numerical problem, you can check the answer by calculating the value of the problem:

       = 160.9968944 . . .

and compare it to the decimal value of the answer that you obtained:

                = 160.9968944 . . .

48.

Remember, you multiply the numbers that are OUTSIDE the radical together, and you keep them OUTSIDE the radical.  Then you multiply the numbers that are INSIDE the radical together and keep them INSIDE the radical. 

However, if you use a calculator and multiply out the numbers that are INSIDE the radical, you end up with a large number that you won’t know how to simplify.  It’s better, instead of multiplying the numbers out, to break them down into prime factors, and for square roots, look for pairs of numbers, for cube roots, look for three of a kind, etc.

Notice that you have three factors of 5!  That makes a perfect cube:

Now, separate into two radicals, with the perfect cube in the first, and the leftover factors in the second radical.

          FINAL ANSWER!!

Since this is a numerical problem, it is a be a good place to check the answer by calculating the problem:

      = 236.9996855 . . .

and then calculate the answer that you obtained:

     = 236.9996855 . . .

(Note:  as I was working this problem for you, I made a mistake and I missed it!  Because of this check, I had to go back and find an error of my own!!)

51.

This is a problem that uses the Distributive Property.  You must multiply the and the .  As always, you multiply the numbers that are OUTSIDE the radical together, and you keep them OUTSIDE the radical.  Then you multiply the numbers that are INSIDE the radical together and keep them INSIDE the radical.

Now, simplify the  (break it down into 4 times 15),

          and the  (break it down into 4 times 5).

       FINAL ANSWER!

Since this is a numerical problem, you might want to check the answer by calculating the value of the problem:

             = 16.60420416 . . .

and the value of the answer that you obtained:

                       = 16.60420416 . . .

Note:  Be careful to close parentheses if your calculator opens them!

52.

This is a problem that uses the Distributive Property.  You must multiply the and the .  As always, you multiply the numbers that are OUTSIDE the radical together, and you keep them OUTSIDE the radical.  Then you multiply the numbers that are INSIDE the radical together and keep them INSIDE the radical.

Now, simplify the  (break it down into 9 times 2),

          and the  (break it down into 4 times 3).

       FINAL ANSWER!

Since this is a numerical problem, you might want to check the answer by calculating the value of the problem:

       = 68.58214165 . . .

and the value of the answer that you obtained:

                = 68.58214165. . .

Note:  Be careful to close parentheses if your calculator opens them!

55.   

            F       O         I        L

As a check, calculate the value of the problem:   =   209.5755077

           Then, calculate the value of the answer:               =    209.5755077

Note:  Be careful to close parentheses if your calculator opens them!

58.   

                    F        O        I         L

 As a check, calculate the value of the problem:   =   -150.6217783

           Then,  calculate the value of the answer:                           =   -150.6217783

Note:  Be careful to close parentheses if your calculator opens them!

60.

         F        O         I         L

As a check, calculate the value of the problem:         =   284.5640646 . . .

Then, calculate the value of the answer:                     =   284.5640646 . . .

Note:  Be careful to close parentheses if your calculator opens them!

64. 

Multiply the first (5) times everything in the second parentheses:

Next, multiply the second  () times everything in the second parentheses:

Now, put it ALL together and combine like terms:

   =  

   =  

   =    

   =                   

Note:  Check by using your calculator to calculate the value of the original problem to see if it comes out to 130.  Be careful to close parentheses if your calculator opens them!

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