Lesson Rationalizing Binomial Denominators in Living Color

1.06  Radicals and Fractional Exponents

Rationalizing Binomial Denominators

College Algebra: One Step at a Time,  Page 80-85:   #3, 5, 6, 8, 9, 15, 18.

Dr. Robert J. Rapalje

Seminole Community College

Sanford, FL  32773

p. 80.  # 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. 81.  # 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. 81.  # 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. 81.  # 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. 81.  # 9.   

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