 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that a. write each expression in radical form 1. 2 y whole waste 2 part 3 upon 4 2. a b whole waste 2 part 2 by 3 and part b says write the following expression in rational exponent form and then simplify it and the expression given is 4th root of 16 into x raised to power 8 into y raised to power 16 the whole Let us start with the solution of the given question. In the a part of the question we have to write each expression in radical form We know that for any real number a and positive integers m and n with n greater than 1 we have a raised to power m upon n is equal to nth root of a whole waste to power m is equal to nth root of a raised to power m So here we have 2 y whole waste 2 part 3 by 4 Now comparing it with a raised to power m upon n we have a is equal to 2 y m is equal to 3 and n is equal to 4 So using this property 2 y whole waste 2 part 3 by 4 can be written as 4th root of 2 y whole cube Now this is equal to 4th root of 2 cube into y cube by using product power rule of exponents which says a b whole waste to power m is equal to a raised to power m into b raised to power m and this is equal to 4th root of 8 into y cube So 2 y whole waste 2 part 3 by 4 is equal to 4th root of 8 into y cube that is 4th root of 8 y cube That is we have written the given expression in radical form Next we have a b whole waste 2 part 2 by 3 Again we have to write this expression in radical form Now again using the same property that is for any real number a and positive integers m and n with n greater than 1 We have a raised to power m upon n is equal to nth root of a whole waste 2 power m is equal to nth root of a raised to power m So using this property a b whole waste 2 part 2 by 3 can be written as cube root of a b whole square And again by using power product rule this is equal to cube root of a square b square So we say that a b whole waste 2 part 2 by 3 is equal to cube root of a square b square That is we have written the given expression in radical form Now we move on to the b part of the question Here we are given an expression that is 4th root of 16 into x raised to power 8 into y raised to power 16 Here we have to write this radical expression in rational exponent form and then we have to simplify it We know that if a is any real number and n is a positive integer where n is greater than 1 Then nth root of a is equal to a raised to power 1 by n Here we restrict a so that a is greater than equal to 0 when n is even Here we are given the expression 4th root of 16 into x raised to power 8 by raised to power 16 Now using this property it can be written as 16 into x raised to power 8 y raised to power 16 whole raised to power 1 by 4 Now using product power rule that is x y whole raised to power m is equal to x raised to power m into y raised to power m where x and y are non-zero real numbers and m is a rational number We have 16 raised to power 1 by 4 into x raised to power 8 whole raised to power 1 by 4 into y raised to power 16 whole raised to power 1 by 4 Here we have now 16 can be written as 2 raised to power 4 whole raised to power 1 by 4 into x raised to power 8 into 1 by 4 into y raised to power 16 into 1 by 4 Now this is equal to 2 raised to power 4 into 1 by 4 that is equal to 1 and 2 raised to power 1 is 2 into x raised to power 2 that is x square into y raised to power 4 So we say that 4th root of 16 into x raised to power 8 by raised to power 16 is equal to 2 into x square by raised to power 4 Hence we have written the given radical expression in rational exponent form which is the required answer to the b part of the question This completes our session hope you enjoyed this session