 Hello and welcome to the session. In this session, we will discuss the following question and the question says simplify by rationalizing the denominator. Part a is 1 upon 3 minus root 5, part b, root 7 upon root 6 plus root 2. Let's start the solution now. In part a, we are given the irrational number 1 upon 3 minus root 5. We have to simplify this irrational number by rationalizing the denominator. To rationalize the denominator, we multiply both numerator and denominator by the rationalizing factor of the denominator. Now the denominator is 3 minus root 5, so it's rationalizing factor is 3 plus root 5, so multiplying numerator and denominator by 3 plus root 5, we get the given irrational number that is 1 upon 3 minus root 5 becomes 1 upon 3 minus root 5 multiplied by 3 plus root 5 upon 3 plus root 5 because 3 plus root 5 is the rationalizing factor. Now this is equal to the numerator is 3 plus root 5 divided by 3 minus root 5 whole multiplied by 3 plus root 5. Now this is equal to 3 plus root 5 divided by 3 square minus root 5 square. Since we know that a plus b whole multiplied by a minus b is equal to a square minus b square. This is equal to 3 plus root 5 divided by 3 square that is 9 minus root 5 whole square that is 5 which is equal to 3 plus root 5 divided by 4. So after rationalizing the denominator, the given irrational number becomes 3 plus root 5 divided by 4 which is our answer for the first part. Now in part b, we are given the irrational number root 7 upon root 6 plus root 2. Now to rationalize the denominator of this irrational number, we will multiply the numerator and the denominator by the rationalizing factor of the denominator. The rationalizing factor of the denominator is root 6 minus root 2. So multiplying numerator and denominator by root 6 minus root 2, we get the given irrational number that is root 7 upon root 6 plus root 2 becomes root 7 upon root 6 plus root 2 multiplied by root 6 minus root 2 whole divided by root 6 minus root 2 because root 6 minus root 2 is the rationalizing factor. So we get this is equal to root 7 whole multiplied by root 6 minus root 2 whole divided by root 6 plus root 2 whole multiplied by root 6 minus root 2. This is equal to the numerator becomes root 7 into root 6 which is equal to root 42 minus root 7 into root 2 which is root 14 whole divided by root 6 whole square minus root 2 whole square which is equal to under root 42 minus under root 14 whole divided by root 6 whole square that is 6 minus root 2 whole square that is 2 which is equal to root 42 minus root 14 whole divided by 4. So after rationalizing the denominator, the given irrational number becomes under root 42 minus under root 14 whole divided by 4. This is our answer for part b. With this we end our session.