 Hello and welcome to the session. In this session we will discuss how to simplify a complex rational expression. Now first of all we will see what a complex rational expression is. A complex rational expression is a rational expression whose numerator and on denominator is also a rational expression. Now x square plus 9 by x square plus 2x plus 1 whole upon 2x minus 6 by 4x plus 2 is an example of the complex rational expression. Now let us see how to simplify a complex rational expression. Now we have two methods to simplify a complex rational expression and in method 1 first we multiply the numerator and denominator of the overall complex fractions by the least common denominator of smaller fractions and then we factorize the numerator and denominator and divide out any common factors. Now let us consider an example, simplify this complex rational expression that is 3 upon x minus 3 minus 4 upon x plus 3 whole upon 7 upon x square minus 9 minus 4 upon x minus 3. Now we know that our first step is to multiply the numerator and denominator of the overall complex fractions by the least common denominator of smaller fractions. Now the denominators of the numerator's fraction have the factors x minus 3 and x plus 3 similarly the denominators of the denominator's fraction have factors x square minus 9 which can be written as x square minus 3 square and that is equal to x plus 3 the whole into x minus 3 the whole by using the formula of a square minus b square which is equal to a plus b the whole into a minus b the whole. Now putting all the different factors together and using the highest exponent we get least common denominator for all the small fractions and that is x minus 3 the whole into x plus 3 the whole. Now multiplying the numerator and denominator by the least common denominator we get 3 upon x minus 3 minus 4 upon x plus 3 the whole into x minus 3 the whole into x plus 3 the whole whole upon 7 upon x square minus 9 minus 4 upon x minus 3 the whole into x minus 3 the whole into x plus 3 the whole and this is equal to 3 upon x minus 3 into x minus 3 the whole into x plus 3 the whole minus 4 upon x plus 3 into x minus 3 the whole into x plus 3 the whole whole upon 7 upon x square minus 9 into x minus 3 the whole into x plus 3 the whole minus 4 upon x minus 3 into x minus 3 the whole into x plus 3 the whole. Now this is equal to 3 into x plus 3 the whole minus 4 into x minus 3 the whole whole upon now 7 upon now x square minus 9 can be written as x plus 3 the whole into x minus 3 the whole into x minus 3 the whole into x plus 3 the whole minus 4 into x plus 3 the whole. the whole. Now this is equal to 3 into x is 3x. 3 into 3 is 9, so we have 3x plus 9. Now minus 4 into x is minus 4x, minus 4 into minus 3 is plus 12. Whole upon 7 minus 4 into x is 4x, minus 4 into 3 is minus 12. Now this is equal to 3x minus 4x is minus x, plus 9 plus 12 will be plus 21. Whole upon, now 7 minus 12 is minus 5, minus 4x or we can also write it as minus x plus 21 whole upon minus 4x minus 5. Hence we have simplified the given rational expression. Now we are going to discuss the second method. In this method first we simplify the numerator and denominator of the complex rational expression such that each only has one rational expression. Then we rewrite the division as multiplication p upon q whole upon r upon s can be written as p upon q divided by r upon s and this is equal to p upon q into s upon r that is p into s upon q into r and then we factorize the numerator and denominator and simplify. Let us consider an example 3 upon a plus 2 upon b whole upon 3 upon b minus 4 upon a square. Let us start with a solution. Now here in the first step we simplify the numerator and denominator of this complex rational expression such that each only has one rational expression. Combining the numerator we get 3 upon a plus 2 upon b will be equal to now here we will get 3b plus 2a in the numerator and in the denominator we will have ab. Now combining the denominator we get 3 upon b minus 4 upon a square will be equal to now in the numerator we will have 3a square minus 4b and in the denominator we have a square b. Now putting these back into the complex fraction we get 3b plus 2a whole upon ab whole upon 3a square minus 4b whole upon a square b. Now rewriting this division as multiplication we get 3b plus 2a whole upon ab into a square b upon 3a square minus 4b and this is equal to here b cancels with b and this a cancels with 1a here. So we are left with a into 3b plus 2a the whole whole upon 3a square minus 4b. Hence we have simplified this complex rational expression by using this second method. Thus in this session we have learned how to simplify a complex rational expression. This completes our session. Hope you enjoyed this session.