Rationalizing Denominators

Intermediate Algebra: One Step at a Time, Pages 259 - 262:   #15, 16, 26.                                                                                                                  Pages 264 - 266:   #3, 5, 6, 8, 11, 12, 14, 20, 21, 22.

 

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

 

 

Pages 259 - 262:  Rationalizing Monomial Denominators

There are three steps in the problems on this page:

            STEP 1:  Simplify the radical.

            STEP 2:  Rationalize the denominator.

            STEP 3:  Reduce the fraction.

p. 261.  # 15. 

SOLUTION. 

STEP 1:  Simplify the radical.

                       

                       

                       

 STEP 2:  Rationalize the denominator.

Multiply the numerator and denominator by

                       

                       

                              

STEP 3:  Reduce the fraction.

Reduce the fraction by dividing out the .

                                  

                       

 

p. 261.  # 16.      

SOLUTION. 

STEP 1:  Simplify the radical.

                       

                       

                       

STEP 2:  Rationalize the denominator.

Multiply the numerator and denominator by

                       

                       

                              

STEP 3:  Reduce the fraction.

Reduce the fraction by dividing out the .

                           

                      

                        

p. 262.  # 26.       

SOLUTION. 

 

If the denominator is a cube root, you need to find a number that you can multiply times this radicand that will make it a perfect cube.   Since you have a factor of 7 in the radical, remember, it will take three of a kind to make a perfect cube.  This means that, since you already have one factor of 7,  you need TWO more factors of 7 to make it a perfect cube.  In other words, multiply numerator and denominator by or

                       

                       

                       

                       

 

 

Pages 263 - 266:  Rationalizing Binomial Denominators

p. 263.  # 3.         

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                       

It is best to multiply out (FOIL) the denominator, but it’s a good idea to leave the numerator in factored form.

                       

The middle term subtracts out, and the 5 – 50 equals -45                  

                          

This reduces by dividing out the 15 with the -45:

             

The tradition in math is to avoid negative denominators, so let’s multiply numerator and denominator times -1. 

          .

                or  

 

p. 264.  # 5.       

Before beginning this problem, notice that you could factor the denominator and reduce the fraction.  It turns out that this step is well worthwhile.

                               Reduce the fraction by dividing out the 12 and the 2.

                               

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                       

It is best to multiply out (FOIL) the denominator, but it’s a good idea to leave the numerator in factored form.

                       

The middle term subtracts out, and the 4 – 3 equals 1

                       

                           or        or    .

 

p. 264.  # 6.                   

Before beginning this problem, notice that you could factor the denominator and reduce the fraction.  It turns out that this step is well worthwhile.

                               Reduce the fraction by dividing out the 12 and the 3.

                               

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                       

It is best to multiply out (FOIL) the denominator, but it’s a good idea to leave the numerator in factored form.

                       

The middle term subtracts out, and the 4 – 3 equals 1

                              or        or    

p. 264.  # 8.               

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                       

 It is best to multiply out (FOIL) the denominator, but it’s a good idea to leave the numerator in factored form.

                           The middle term always subtracts out!                  

                                Simplify the denominator

                                 Divide out the 6 and -30.

                                   Avoid negative denominators, so multiply numerator and denominator by -1.

   or   

 

p. 265.   #11.      

 

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                              

 

It is best to multiply out (FOIL) the denominator, and since there are also radicals in the numerator, it will be necessary to multiply out the numerator as well.  It might be helpful to go ahead and simplify

 

                          The middle term always subtracts out!

                       

                                Simplify the numerator and denominator.

 

                         

                         

                             

                       

                       

Since this is a numerical problem, you can check with your calculator by calculating the decimal approximation of the problem and of the answer to see if they agree.

         =  1.259575563                    

 =   1.259575563        It checks!!    

 

p. 265.   #12.      

 

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                             

 

It is best to multiply out (FOIL) the denominator, and since there are also radicals in the numerator, it will be necessary to multiply out the numerator as well.  It might be helpful to go ahead and simplify

 

                          The middle term always subtracts out!

                       

                                Simplify the numerator and denominator.

 

                         

                         

                             

                       

                       

Since this is a numerical problem, you can check with your calculator by calculating the decimal approximation of the problem and of the answer to see if they agree.

         =  0.5292788193      

 =   0.5292788193        It checks!!    

 

p. 265.  # 14.               

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                       

It is best to multiply out (FOIL) the denominator, and since there are also radicals in the numerator, it will be necessary to multiply out the numerator as well.  It might be helpful to go ahead and simplify

                          The middle term always subtracts out!               

                                   Simplify the numerator and denominator.

                         

                         

                                              

                       

                          or 

                                   

p. 266.  # 20.              

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                         

In this case, you must multiply out (F OI L) both the numerator and the denominator.

                             Simplify numerator and denominator.                    

                                          

                                          

                                          Factor common factor of 3 from the numerator in order to reduce the fraction.                    

                                     Since the 3's are factors, you can divide them out!

                             

This is a great place to check with the calculator!  Use your calculator to calculate the problem.  Be sure to put parentheses around the numerator and denominator, and if your calculator opens parentheses when you enter square root, be sure to close the parentheses after you enter the number! 

Problem =  should give you   .1715728753 . . .  

Answer =  also should give you   .1715728753 . . . 

This checks!!                       

 

p. 266.   #21.      

 

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

           

 

In this case, you must multiply out (F OI L) both the numerator and the denominator.

 

              Simplify numerator and denominator.

                       

                                        

 

                         

                              Factor the numerator in order to reduce the fraction.

                       

                              Divide out the -2 and the 10.

                       

                 or   

Finally, as a check, if you calculate the decimal approximation of the problem and of the answer, you should get approximately -69.78505. 

 

p. 266.  # 22.                 

Rationalize the denominator by multiplying numerator and denominator of the fraction by the conjugate of the denominator.  That is the same as the denominator by with opposite sign:  .

                         

In this case, you must multiply out (F OI L) both the numerator and the denominator.

                            Simplify numerator and denominator.        

                                        

                            

                            

                                   Factor the numerator in order to reduce the fraction.                       

                                 Divide out the 2 and the 10.

                                                  

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Dr. Robert J. Rapalje Altamonte Springs Campus
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