Comple ational Epressions What is a Comple ational Epression? A omple rational epression is similar to a omple fration whih has one or more frations in its numerator, denominator, or both The following are all eamples of omple rational epressions Simplifying ational Epressions: There are two methods used to simplify a rational epression: the Division Method, and the LCD method We will look at both methods in the following eamples The Division Method: This method follows quite naturally from work with arithmeti frations It is based on the idea that a omple fration is really just a division problem Here is the general proedure for the division method Write the numerator and denominator as a single rational epression in simplified form ewrite the epression as a division problem Divide and simplify if possible Eample: Simplify Solution: Sine the numerator and denominator are already single frations, begin by rewriting the problem as a division problem, and then simplify aordingly ewrite the omple fration as a division problem ewrite division as a multipliation problem Multiply and simplify if possible
Eample: Simplify Solution: In this problem the numerator onsists of two rational epressions and the denominator onsists of two rational epressions We need to rewrite the numerator and denominator as one rational epression Then proeed as in the previous eample Add the frations in the numerator and denominator to reate a single fration in eah ewrite the omple fration as a division problem ewrite division as a multipliation problem Multiply and simplify if possible Eample: Simplify Solution: Add the frations in the numerator and denominator to reate a single fration in eah ewrite the omple fration as a division problem ewrite division as a multipliation problem Multiply and simplify if possible ) )
Eample: Simplify Solution: Add the frations in the numerator and denominator to reate a single fration in eah Simplify the numerator and denominator ewrite the omple fration as a division problem ewrite division as a multipliation problem Multiply Fator & Simplify if possible Eample: Simplify Solution: Add the frations in the numerator and denominator to reate a single fration in eah ewrite the omple fration as a division problem ewrite division as a multipliation problem Multiply, fator and simplify if possible ) ) ) ) ) ) )
The LCD Method: This method requires us to find the Least Common Denominator LCD) of all rational epressions in the problem It is atually a muh simpler method Here is the general proedure Find the LCD of all the rational epressions within both the numerator and denominator Multiply the numerator and denominator of the omple rational epression by the LCD Simplify if possible Eample: Simplify Solution: The LCD is 0 ) ) Eample: Simplify Solution: The LCD is ) ) 8 Eample: Simplify Solution: The LCD is ) )
Eample: Simplify Solution: The LCD is ) ) ) ) ) ) ) ) ) Eample: Simplify Solution: The LCD is ) ) 7 ) ) ) )
7-Appliations Eample: An epression from the study of eletriity is V where V represents voltage and represents resistane Simplify this epression Solution: I am going to use the LCD method The LCD is ) ) ) ) ) ) V V V Eample: The epression below is important in the design of airplanes Write this epression in simplified form p m Solution: One again, I am going to use the LCD method The LCD is p m p m Eample: On a trip, an airplane flew at an average speed of a mph, returning on the same route at an average speed of b mph The planes average speed for the round trip is given by the omple rational epression b a Simplify this omple rational epression Solution: In this ase I am going to use the division method a b ab ab b a ab ) )