 Hello and welcome to the session. In this session we shall discuss how to write radical expressions as an expression with a rational exponent and vice versa. In our earlier session we have already discussed radical notation that is n is root of x where n is index and x is called radicant. In this session we can use this concept to develop a new notation using exponents. This new notation involves rational numbers as exponents. We know that rational numbers are of the form p by q where q is not equal to 0 and p and q are constants. Let us start with an example. Suppose we are given that a is equal to 9 raised to power 1 by 2. Let us name this equation as equation number one. Now squaring both sides of equation number one we get a square is equal to 9 raised to power 1 by 2 whole square which implies that a square is equal to 9. Let us name this as equation number two. From equation number two we can see that a is the number whose square is 9 that is a is the square root of 9. So we have a is equal to square root of 9 and let us name this equation as equation number three. Now from equation number one and equation number three we see that 9 raised to power 1 by 2 is equal to square root of 9. So we conclude the definition of rational exponents as if a is any real number and n is a positive integer that is n is greater than 1 then a raised to power 1 by n is equal to n is root of a. Here we restrict a so that a is greater than equal to 0 when n is even. For any real number a and positive integers m and n with n greater than 1 we have a raised to power m by n is equal to nth root of a whole raised to power m and this is also equal to nth root of a raised to power m. These two radical forms of a raised to power m by n are equivalent and we can use any of them depending on whether we are evaluating numerical expressions or rewriting expressions containing variables in radical form. For negative exponents we have a raised to power minus of m by n is equal to 1 upon a raised to power m by n. All the properties of exponents which we had discussed earlier will also hold for rational exponents. Now we shall see rules and properties of exponents for any non-zero numbers x and y and rational numbers m and n. Now first is product rule that is x raised to power m into x raised to power n will be equal to x raised to power m plus n. Similarly we have quotient rule that is x raised to power m upon x raised to power n is equal to x raised to power m minus n. Next is power rule that is x raised to power m whole raised to power n will be equal to x raised to power m into n that is mn. Next is product power rule that is x y whole raised to power m will be equal to x raised to power m into y raised to power m. Next is quotient power rule that is x upon y whole raised to power m will be equal to x raised to power m upon y raised to power m. Here it should be noted that we restrict x and y so that x and y are greater than equal to 0 when m and n indicate even roots. Now here let us discuss an example. Simplify 16 upon 625 whole raised to power 3 by 4. Now using this property of rational exponent we can write 16 upon 625 whole raised to power 3 by 4 as 16 upon 625 whole raised to power 1 by 4 whole cube. Here we have used this property that is a raised to power m whole raised to power n is equal to a raised to power mn. Now using this property a raised to power 1 by n is equal to nth root of a we write fourth root of 16 upon 625 whole cube. Now fourth root of 16 upon 625 will be equal to 2 by 5 and this whole cube and this is equal to 2 into 2 into 2 whole upon 5 into 5 into 5 which is equal to 8 upon 125. So we have got 16 upon 625 whole raised to power 3 by 4 is equal to 8 upon 125. Let us take another example. Here we have to write the radical expression in exponential form. The given radical expression is cube root of 27a cube b raised to power 9. Now using the definition of rational exponents we can write this radical expression as 27a cube b raised to power 9 whole raised to power 1 by 3. Now using power product rule we have 27 raised to power 1 by 3 into a cube whole raised to power 1 by 3 into b raised to power 9 whole raised to power 1 by 3. Now here again using power rule we get now here 27 can be written as 3 raised to power 3 whole raised to power 1 by 3 into a raised to power 3 into 1 by 3 that is 1 into b raised to power 9 into 1 by 3 that is 3. So this is equal to 3 raised to power 3 into 1 by 3 that is 1. So we have 3 into a into b raised to power 3 that is 3a b cube. So we say that the exponential form of the given radical expression is 3a b cube. Thus in this session we have discussed how to write radical expressions as an expression with a radical exponent and vice versa. This completes our session. Hope you enjoyed this session.