 Hello and welcome to the session. In this session we are going to discuss the following question. And the question says that, use rational exponents to write this expression as single radical expression. And the expression given is, cube root of x into fourth root of 2x. Now let us start with the solution of the given question. In this question we are given an expression that is cube root of x into fourth root of 2x. Now using radical exponents we have to express it as single radical expression. We know that 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 nth root of a. So here cube root of x can be written as x raised to power 1 by 3 and fourth root of 2x can be written as 2x whole raised to power 1 by 4. Now using power product law that is xy whole raised to power m is equal to x raised to power m into y raised to power m. Thus we have 2x whole raised to power 1 by 4 is equal to 2 raised to power 1 by 4 into x raised to power 1 by 4. So we can write cube root of x into fourth root of 2x is equal to x raised to power 1 by 3 into 2 raised to power 1 by 4 into x raised to power 1 by 4. Now this can be written as 2 raised to power 1 by 4 into x raised to power 1 by 3 into x raised to power 1 by 4. Now using product law of exponents we have x raised to power m into x raised to power n is equal to x raised to power m plus n. So here we get 2 raised to power 1 by 4 into x raised to power 1 by 3 plus 1 by 4 and this is equal to 2 raised to power 1 by 4 into x raised to power. Now taking LCM and solving here we get the power as 7 by 12. So here we get 2 raised to power 1 by 4 into x raised to power 7 by 12 and we can rewrite this as 2 raised to power 3 into 1 by 12 into x raised to power 7 into 1 by 12. Now again using power product law this can be written as 2 cube into x raised to power 7 whole raised to power 1 by 12. So cube root of x into 4th root of 2x is equal to 2 cube into x raised to power 7 whole raised to power 1 by 12 and this is equal to 12th root of 2 cube into x raised to power 7 Using this property that is a raised to power 1 by n is equal to nth root of a on solving further we get 12th root of now 2 cube is 8 into x raised to power 7. So we have 8 into x raised to power 7 that is 8x raised to power 7 that is 12th root of 8x raised to power 7 so we have written the given expression as single radical expression that is 12th root of 8 into x raised to power 7 which is the required answer. This completes our session hope you enjoyed this session.