 Hello and welcome to this session. In this session, we will discuss operations on real numbers. Rational numbers are closed with respect to addition, subtraction, multiplication and division. That is, when we add, subtract, multiply and divide two rational numbers, we get a rational number itself. However, the sum, difference, quotients and products of irrational numbers are not always irrational. Consider an irrational number root 3. When we multiply root 3 with the rational number root 3 itself, we get 3 which is not an irrational number. Now we have some results which says that sum or difference of a rational number and an irrational number is irrational. Then the product or the quotient of a non-zero rational number with an irrational number is irrational. And then we have that if we add, subtract, multiply or divide two irrationals, the result may be rational or irrational. Let's take a rational number a, then irrational number root b. Then according to the above results, we have that a plus root b, a minus root b, a into root b and a upon root b would all be irrational numbers. Consider positive real numbers a and b. Then we have the following identities. According to which we have square root ab is equal to square root a into square root b. Then the next identity is square root a upon b is equal to square root a upon square root b. Next one is square root a plus square root b multiplied with square root a minus square root b is equal to a minus b. Then we have a plus square root b multiplied with a minus square root b is equal to a square minus b. And the next identity is square root a plus square root b, the whole square is equal to a plus 2 into root ab plus b. Also we have one more identity, square root a plus square root b multiplied with square root c plus square root d is equal to square root ac plus square root ad plus square root bc plus square root bd. Let's consider an example, square root 13 minus square root 6 multiplied with square root 13 plus square root 6. To evaluate this we will use this identity. So on putting a equal to 13 and b equal to 6 we get this equal to 13 minus 6 which is equal to 7. This is how we can use different identities to simplify the given expressions. When the denominator of an expression contains a term with a square root the process of converting it to an equivalent expression whose denominator is a rational number is called rationalizing the denominator. So we have to rationalize the denominator of 1 upon square root a plus b. We multiply this by square root a minus b upon square root a minus b where we have a and b are integers. Consider 1 upon square root 3 plus 4. Let's rationalize the denominator of this expression. To do this we multiply 1 upon square root 3 plus 4 by square root 3 minus 4 upon square root 3 minus 4. So this comes out to be equal to square root 3 minus 4 upon 3 minus 16 that is we get this is equal to square root 3 minus 4 upon minus 13 or we can also write it as 4 minus square root 3 upon 13. Now we discuss loss of exponents for real numbers. Let a greater than 0 be a real number and p and q be rational numbers. Then we have a to the power p multiplied by a to the power q is equal to a to the power p plus q. Then a to the power p whole to the power q is equal to a to the power pq. Next is a to the power p upon a to the power q is equal to a to the power p minus q. Then we have a to the power p multiplied by b to the power p is equal to a b whole to the power p. Consider the expression 5 to the power 1 upon 3 upon 5 to the power 1 upon 6. Let's simplify this expression. For this, we will use this law of exponent which says, a to the power p upon a to the power q is equal to a to the power p minus q. So, here we will put p equal to 1 upon 3 and q equal to 1 upon 6. So we get this is equal to 5 to the power 1 upon 3 minus 1 upon 6, which comes out to be equal to 5 to the power 1 upon 6. So this is how we use different laws of exponents for real numbers to simplify the given expressions. This completes the session. Hope you understood the concept of operations on real numbers.