Complex Fractions --Part II

Intermediate Algebra: One Step at a Time.  Pages 197-200:   #29, 30, 31, 33, 34, 36, 39, 40, 41, 43, 47, 49, 51,  53, extras.

College Algebra: One Step at a Time.    Pages 59-62:    #29, 30, 31, 33, 34, 36, 39, 40, 41, 43, 47, 49, 51, 53, extras.

 

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

 

For additional exercises, please see Complex Fractions Part I.

 

29.  Method I:        =          

 

 

The LCD for the first (red) part is x(x+2),  for the second (blue) part is also x(x+2),   so multiply numerator and denominator of each fraction by the appropriate missing factors:

                            =                                                         

                             =          

Put each of these LCDs in place:

                             =        

Multiply out the numerators:

                              =              

Combine like terms for each of the numerators:

                              =              

Factor the numerators (if possible!), invert the second fraction, and multiply:

                              =                

Divide out factors of any numerator with corresponding factors from the denominators.  In particular, divide out the factors of x, (x+2), and 2:

                               =                

 

 

30.  Method I:

    =          

 

The LCD for the first (red) part is x(x−2),  for the second (blue) part is also x(x−2),   so multiply numerator and denominator of each fraction by the appropriate missing factors:

                            =                                                          

                             =          

Put each of these LCDs in place:

                             =        

 Multiply out the numerators:

                              =              

 Combine like terms for each of the numerators:

                               =             

Factor the numerators (if possible!), invert the second fraction, and multiply:

                              =               

Divide out factors of any numerator with corresponding factors from the denominators.  In particular, divide out the factors of x, (x−2), and 2:

                                =                  or  or

 

31.  Method I:         =         

 

 

The LCD for the first (red) part is x(x+2),  for the second (blue) part is x(x−2),   so multiply numerator and denominator of each fraction by the appropriate missing factors:

                            =                                                         

                            =          

Put each of these LCDs in place:

                             =        

Multiply out the numerators:

                              =             

Combine like terms for each of the numerators:

                              =             

Factor the numerators (if possible!), invert the second fraction, and multiply:

                              =               

Divide out factors of any numerator with corresponding factors from the denominators.  In particular, divide out the factors of  x  and  2:

                               =                   also correct:

33.  Method I:            =          

 

 

 

The LCD for the first (red) part is x(x−4),  for the second (blue) part is x(x+1),   so multiply numerator and denominator of each fraction by the appropriate missing factors:

                            =          

                            =                            

 

Put each of these LCDs in place:

                             =        

 

Multiply out the numerators:

                              =              

 

Combine like terms for each of the numerators:

 

                              =             

Factor the numerators (if possible!), invert the second fraction, and multiply:

                              =               

Divide out factors of any numerator with corresponding factors from the denominators.  In particular, divide out the factors of  x , 2, and  the  x+4.  The final answer is

 

                               =                   

 

 

 

 

34.  Method I:          =          

 

 

 

The LCD for the first (red) part is x(x+y),  for the second (blue) part is x(x−y),   so multiply numerator and denominator of each fraction by the appropriate missing factors:

                            =          

                            =                          

 

Put each of these LCDs in place:

                             =        

 

Multiply out the numerators:

                              =              

 

Combine like terms for each of the numerators:

 

                              =             

Factor the numerators (if possible!), invert the second fraction, and multiply:

                              =             

Divide out factors of any numerator with corresponding factors from the denominators.  In particular, divide out the factors of  x  and  2:

 

                               =                   

 

 

36.           =             

The LCD for the whole problem is 2xy, so multiply numerator and denominator by 

 Method II:                    =

 

                                         = 

 

Although this looks terrible, it is really quite simple, and you can do it in your head.  Normally, you don’t even write down the previous step.  It all simplifies to this:

                                         = 

 

39.           =  

                                       = 

The LCD for the whole problem is 16x2.  Multiply numerator and denominator by .

Method II:                  =    

 

Next, use the distributive property.  You may want not need to show your work in the next step, but rather just multiply and simplify the fractions in your head. 

                                      =  

It all simplifies down to this:

                                      = 

 40.         =  

The LCD for the whole problem is x2y2.  Multiply numerator and denominator by .

 Method II:   =     

 

Next, use the distributive property.  You may want not need to show your work in the next step, but rather just multiply and simplify the fractions in your head. 

                      =  

It all simplifies down to this:

                      = 

which factors into this:

                      =   

and reduces (divide out the ( y − x) factor!) to this:

                      =     Final answer!

 

 41.

  =  

The LCD for the whole problem is 2x.  Method II is appropriate, so you should multiply numerator and denominator by .

                        =    

 

Next, use the distributive property.  You may want not need to show your work in the next step, but rather just multiply and simplify the fractions in your head. 

                       =  

 It all simplifies down to this:

                       = 

which factors into this:

                       =   

and reduces (divide out the ( 2x + 1) factor!) to this:

                      =   

 

 43.

  =  

The LCD for the whole problem is 4x.  Multiply numerator and denominator by .

                       =    

 

Next, use the distributive property.  You may want not need to show your work in the next step, but rather just multiply and simplify the fractions in your head. 

                      =  

It all simplifies down to this:

                      = 

which factors into this:

                      =    or  

and reduces (divide out the ( x − 2) factor!) to this:

                      =   

47.      Short cuts don’t work here.  Back to basics  and .

        You must have a single fraction, so find the LCD = .

           A single fraction raised to the -1 power means to invert the fraction.         

                  Final answer, unless you prefer to factor the denominator:

 

 49.     Short cuts don’t work here.  Back to basics  and .

 

        You must have a single fraction, so find the LCD = .

 

           A single fraction raised to the -2 power means to invert the fraction and square it.

          Invert the fraction:  and square the numerator and denominator: 

 

NOTE:  You could write it , but do NOT write   !!

 

 

51.      Short cuts don’t work here.  Back to basics  and .

 

        You must have a single fraction, so find the LCD = .

 

           A single fraction raised to the -1 power means to invert the fraction.

            Final answer, unless you prefer to factor out the 3:

 

 53.    

                    LCD = 3xy

          The first fraction is “missing” y; second missing the x.

          The LCD is 3xy, and the numerator is y − x

                        The last step is to invert the fraction!

                               Final Answer!!

 

EXTRA PROBLEM:       Simplify the fraction:  

When you eliminate the negative exponent, a complex fraction results.

             =  

Use the “unstacking method”, rewriting the fraction as the red fraction divided by the blue fraction.

                         = 

                        =   

                        =  

Invert the second fraction and multiply.  Divide out the (1+ 2x) factor:

                        =   

                        =      

                        =            

 

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Dr. Robert J. Rapalje Altamonte Springs Campus
Contact me at:   rapaljer@seminolestate.edu
Phone number:  NONE Retired!!
OFFICE:          NONE  
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