 Hello and welcome to the session. In this session we will learn about rationalization f in rational numbers. In rational numbers root 2 root 3 and so on are also called radicals or cells. As they cannot be written as squares of any rational number. Now let us discuss some properties, operations and radicals. Now property number 1 is addition, fraction or radicals. Now root 3 plus root q is not equal to root p plus q, but root p plus root p is equal to root p. That is, like radicals can be added, like radicals cannot be added. Now the same property will hold subtraction also. root p minus root q is not equal to root p minus q, but root p minus root p is equal to 0. That the like terms can be added or subtracted, but unlike terms cannot be added or subtracted. Now the second property, the multiplication of radicals. Now root p into root q is equal to root p into q or root p q and if a rational number is multiplied then we obtain into root q is equal to p root q. Now the third property, now here root p root q is equal to root p over q. Now let us discuss, we obtain the number, rational number is equal to root 14 which is an irrational number. root 2 into root 32 is equal to root 64 which is equal to 8 is a rational number. Therefore, we obtain rational numbers. For example, we are getting a rational number. Therefore, these two irrational numbers are the factors of each other. A rationalizing factor root 2 is equal to root 16 which is equal to 4 rational numbers and we are getting a rational number. Therefore, these two irrational numbers are rationalizing factors of each other. Now the definition of rationalization is the process of rational number. Now the rationalized rational denominator is when denominator numerator to denominator. Now let us discuss one example for this. Now here solution for this. Now here the rationalized root 3 we will multiply and divide it will be root 3 into root 3 by root 3 root 3 over root 3 into root 3. Now here 3 is a rational number. So we obtain a rational denominator. The denominator multiply the denominators that is q3 and conjugate but we are changing the sign in the middle. Let us discuss one example for this which is rationalize. Now we will start with the solution of this. Now here the conjugate 4 minus root 3. Now multiply root 3 by root over 4 plus root 3 into 4 minus root 3 over 4 minus root 3. Now this is equal to 3 into 4 minus root 3 the whole, whole upon 4 plus root 3 the whole into 4 minus root 3 the whole which is further equal to 12 minus 3 root 3 whole upon. Now using the formula of a plus b the whole into a minus b the whole it will be a square minus b square. So it will be 4 square minus root 3 square which is further equal to 12 minus 3 root 3 whole upon 30. The denominator is 13 which is a rational number therefore we have learned about rationalization of irrational numbers. This completes the session by this session.