we are asked to simplify nine p to the two thirds all of that over the square root of the four p cube so lets just do this step by step the first thing we can do is well if its nine p of the two third thats the same thing as nine to the two third times p to the two third that's what the stop part that we can simplify too now the bottom part we can rewrite the bottom part right here the square root of four p to the third this is the exact same thing as four p to the third to the one half part power this is the exact same thing as four p to the third to the one half power this is the same thing square root of four p to the third is same thing as four p to the third to the one half power so given that we can do the same thing we did on the numerator we can say well this is the same thing as four to the one half four to the the one half times p to the third to the one half power now lets see that we can simplify this even more so this is going to be equal to lets do the denominator first here so whats four to the one half power what's the principle rule of four well two squared is four so its two its positive root is two and then p to the third and that to the one half that's the same thing p to the three times one half or p to the three half p to the three over two nine to the two thirds so the cube root of nine is anything easy well actually yo can think by this way yo could say its nine squared so we can do nine to the two thirds le me do this this way this is the best to think about it this nine to the two thirds one could write it some kind of making it unsimplifying it but it will help calculate it what it actually means nine to the two third is the same thing as nine squared to the one third power right because two times one third is two thirds and you have that off course times p to the two third and this is going to be eqal to nine squared is eighty one so its eighty one to the one third power times p to the two thirds now instead of writing this two and this p to the three half down here let me just write it as negative exponents so you are going to just write it like this the two in the denominater is the same thing as two to the negative one or one half so this right here somewhere in the numerator is the same thing as one half that and that is the the same thing and p to the three half in the denominator is the same thing as p as p to the negative p to the negative three half so this whole expressions so far simplified to this right over here let me copy and past to the next line now so let me copy it and paste bring it down right over there let's see if can simplify this even more so let's think about eighty one if we can write it as well essentially we take something to the cube we take the cube root or something we are taking some we are taking some thing one third power taking the cube root of it so we have to find some numbers or may be we can factor it into things that do have cube root and other things they don't so eighty one is divisible by three its three times twenty seven which is three times nine which is three times three so eighty one is three to the fourth power or its three multiplied by itself three times so we could rewrite this eighty one to the one third we could rewrite as three times three times three to the well alright like this times three to the one third that's the same thing as eighty one and let's think about this ps and i am doing more steps that if want to on piece of paper so i really want you to understand kind of what's going on here so p to the two thirds times p to the negative three half so that's going to be times p to the two thirds two thirds minus three half power so we did the ps then we just had the times one half times one half now this right over here this is the same thing this is the same thing as we could just use this as three times three times three so its the same thing as three times three times three to the one third times three to the one third right we done this multiple time times the two third minus three half so lets just work that out actually three half is the larger so