1.06  Radicals and Fractional Exponents

Rationalizing Monomial Denominators

College Algebra: One Step at a Time,  Page 77-79:   #13, 17, 21, 23, 25, 27.

Dr. Robert J. Rapalje

Seminole Community College

Sanford, FL  32773

13.       

Notice that the denominator has a cube root that is not a perfect cube!  The goal, in rationalizing the denominator, is to get a perfect cube for the denominator.  In this case, it would be nice to get a denominator like.  To do this, you need to multiply numerator and denominator by .  It looks like this:

Divide out the factor of 3:

As a check, calculate the value of the problem:    = 4.160167646

then calculate the value of your answer :             = 4.160167646

17.             

Notice that the denominator has a cube root that is not a perfect cube!  The denominator does factor into .  The goal, in rationalizing the denominator, is to get a perfect cube for the denominator.  In this case, it would be nice to get a denominator like.  To do this, you need to multiply numerator and denominator by .  It looks like this:

Divide out the factor of 7:

As a check, calculate the value of the problem:  = 9.564655914

then calculate the value of your answer :             = 9.564655914

21.             

Notice that the denominator has a cube root that is not a perfect cube!  The denominator does contain 4 which is .  The goal, in rationalizing the denominator, is to get a perfect cube for the denominator.  In this case, it would be nice to get a denominator like.  Since you already have , to get the , you need to multiply numerator and denominator by .  As to the variables , for perfect cubes, it is necessary to get exponents that are divisible by 3, such as .  Since you already have the  , to get the , you will need to multiply numerator and denominator by .  Altogether, it looks like this:

                               

The denominator is (of course!) a perfect cube, so you can simplify this:

The fraction reduces, so divide out the .             

23.             

Notice that the denominator has a cube root that is not a perfect cube!  The denominator does contains 25 which is .  The goal, in rationalizing the denominator, is to get a perfect cube for the denominator.  In this case, it would be nice to get a denominator like.  Since you already have , to get the , you need to multiply numerator and denominator by .  As to the variables , for perfect cubes, it is necessary to get exponents that are divisible by 3, such as .  Since you already have the  , to get the , you will need to multiply numerator and denominator by .  Altogether, it looks like this:

The denominator is (of course!) a perfect cube, so you can simplify this:

The fraction reduces, so divide out the .              

25.              =  

Notice that the denominator has a fifth root that is not a perfect fifth power!  The denominator contains 4 which is .  The goal, in rationalizing this denominator, is to get a perfect fifth power for the denominator.  In this case, it would be nice to get a denominator like.  Since you already have , to get the , you need to multiply by .  As to the variables , for perfect fifth powers, it is necessary to get exponents that are divisible by 5, such as .  Since you already have the  , to get the , you will need to multiply numerator and denominator by .  Altogether, it looks like this:

The denominator is (of course!) a perfect cube, so you can simplify this:

The fraction reduces, so divide out the .               

 27.              =  

Notice that the denominator has a fifth root that is not a perfect fifth power!  The denominator contains 16 which is .  The goal, in rationalizing this denominator, is to get a perfect fifth power for the denominator.  In this case, it would be nice to get a denominator like.  Since you already have , to get the , you need to multiply numerator and denominator by .  As to the variables , for perfect fifth powers, it is necessary to get exponents that are divisible by 5, such as .  Since you already have the  , to get the , you will need to multiply numerator and denominator by .  Altogether, it looks like this:

The denominator is (of course!) a perfect cube, so you can simplify this:

The fraction reduces, so divide out the .              

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