 Hi and how are you all today? I am Priyanka and I will be helping you with the following question which says rationalize the denominator of the following. These are the four parts of the question which are given to us and in each part we need to rationalize the denominator. So let us start with our solution of the first part. The expression is given to us as 1 by root 7. As we know that root 7 when gets multiplied by root 7 we get 7 which is a rational number. So in order to rationalize the denominator we will multiply both the numerator and the denominator by root 7. And we will write it multiply numerator and denominator by root 7. So, proceeding on we have root 7 divided by 7 and this becomes the answer of our first part. Proceeding on with the solution of second part the expression given to us is 1 divided by root 7 minus root 6. Now in order to rationalize we need to multiply the numerator and the denominator we have denoted it by N and D by the conjugate of the denominator. Right, so its conjugate will be root 7 plus root 6. So on doing so we have 1 divided by root 7 minus root 6 getting multiplied by root 7 plus root 6 in the denominator as well as the numerator. So we have root 7 plus 6 divided by using the identity that is a minus b a plus b is equal to a square minus b square which is applicable in the denominator. We can write a square as root 7 the whole square minus root 6 the whole square. Let us simplify it little further. On doing so we have root 7 plus root 6 divided by 7 minus 6 that gives us root 7 plus root 6 by 1. So the answer to the second part is root 7 plus root 6. So this completes the answer to the second part. Proceeding on with the third part the expression is 1 divided by root 5 plus root 2. Here also we will multiply the numerator and the denominator by the conjugate of the denominator that is root 5 minus root 2. On doing so we have 1 divided by root 5 plus root 2 getting multiplied by root 5 minus root 2 to both the sides. Further root 5 minus root 2 and in the denominator we can again apply the identity that is a plus b a minus b is equal to a square minus b square. So we have root 5 the whole square minus root 2 the whole square. On simplifying it little further we have root 5 minus root 2 divided by 5 minus 2 which gives us root 5 minus root 2 by 3. So this becomes the answer to the third part. Proceeding on with the last and final part here also the expression which is given to us is 1 divided by root 7 minus 2 multiplying the numerator and the denominator by the conjugate of the denominator that is root 7 plus 2 we have 1 divided by root 7 minus 2 getting multiplied by root 7 plus 2. Further simplifying it we have root 7 plus 2 divided by root 7 the whole square minus 2 the whole square because we have again applied our knowledge of the identity that is a minus b a plus b is equal to a square minus b square. Now here our a is root 7 and b is 2. So we have root 7 plus 2 divided by 7 minus 4 which gives us root 7 plus 2 divided by 3 and this becomes the answer to the last and final part which was given us. So I hope you enjoyed the session and learned how to rationalize the denominator well. Bye.