Evaluating mixed radicals and exponents

let's see if we can simplify 6 to the 1/2 power times the fifth root of 6 and all of that to the 3rd power and encourage you to pause this video and try it on your own so let me actually color code these exponents just so we can keep track of them a little better so that's the 1/2 power in blue this is the fifth root here in magenta and let's see in green let's think about this third power so one way to think about this fifth root is that this is the exact same thing as raising this six to the 1/5 power so let's write it like that so this part right over here we could rewrite as 6 to the 1/5 power 6 to the 1/5 power and then that whole thing gets raised to the 3rd power that whole thing gets raised to the 3rd power and of course we have this 6 to the 1/2 power out here 6 to the 1/2 power times all of all of this business right over here now what happens if we raise something to an exponent and then raise that whole thing to another exponent well we've already seen in our exponent properties that's the equivalent of raising this to the product of these two exponents so this part right over here could be re-written as this part right over here could be re-written as 6 6 to the 3 times 1/5 is 3/5 6 to the 3/5 power 6 to the 3/5 power and of course we're multiplying that times 6 to the 1/2 power 6 to the 1/2 power times 6 to the 3/5 power and now if you're multiplying some base to summit to this exponent and then the same base again to another exponent we know that this is going to be the same thing and actually we could put these equal signs the whole way because these all equal each other this is the same thing as 6 being raised to the 1/2 plus 3/5 power 1/2 plus 3 over 3 over 5 now what's 1/2 plus 3 over 5 well we could find a common denominator it would be 10 so that's the same as let me just write it this way this is the same thing as 6 to the instead of 1/2 we can write it as 5 over 10 plus 3/5 is the same thing as 6 over 10 - 6 over 10 power which is the same thing and we deserve a little bit of a drumroll here this wasn't that this wasn't that long of a problem 6 to the 11 tenth power 6 to the others right and all 11 over 10th power and so that looks pretty simplified to me I guess we're done