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we are asked to simplify nine p to the two thirds all of that over the square root of the four p cube

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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

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the same thing as nine to the two third times p to the two third that's what the stop part that we can

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simplify too now the bottom part we can rewrite the bottom part right here the square root of four p

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to the third this is the exact same thing as four p to the third to the one half part power this is the

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exact same thing as four p to the third to the one half power this is the same thing square root of four

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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

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same thing we did on the numerator we can say well this is the same thing as four to the one half

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four to the the one half times p to the third to the one half power now lets see that we can simplify

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this even more so this is going to be equal to lets do the denominator first here so whats four to the

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one half power what's the principle rule of four well two squared is four so its two its positive root

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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

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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

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anything easy well actually yo can think by this way yo could say its nine squared so we can do nine

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to the two thirds le me do this this way this is the best to think about it this nine to the two thirds

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one could write it some kind of making it unsimplifying it but it will help calculate it what it actually

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means nine to the two third is the same thing as nine squared to the one third power right because two

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times one third is two thirds and you have that off course times p to the two third and this is going

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to be eqal to nine squared is eighty one so its eighty one to the one third power times p to the two

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thirds now instead of writing this two and this p to the three half down here let me just write it as

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negative exponents so you are going to just write it like this the two in the denominater is the same

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thing as two to the negative one or one half so this right here somewhere in the numerator is the same

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thing as one half that and that is the the same thing and p to the three half in the denominator is the

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same thing as p as p to the negative p to the negative three half so this whole expressions so far simplified

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to this right over here let me copy and past to the next line now so let me copy it and paste bring it

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down right over there let's see if can simplify this even more so let's think about eighty one if we

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can write it as well essentially we take something to the cube we take the cube root or something we

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are taking some we are taking some thing one third power taking the cube root of it so we have to find

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some numbers or may be we can factor it into things that do have cube root and other things they don't

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so eighty one is divisible by three its three times twenty seven which is three times nine which is three

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times three so eighty one is three to the fourth power or its three multiplied by itself three times

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so we could rewrite this eighty one to the one third we could rewrite as three times three times three

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to the well alright like this times three to the one third that's the same thing as eighty one and let's

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think about this ps and i am doing more steps that if want to on piece of paper so i really want you

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to understand kind of what's going on here so p to the two thirds times p to the negative three half

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so that's going to be times p to the two thirds two thirds minus three half power so we did the ps then

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we just had the times one half times one half now this right over here this is the same thing this is

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the same thing as we could just use this as three times three times three so its the same thing as three

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times three times three to the one third times three to the one third right we done this multiple time

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times the two third minus three half so lets just work that out actually three half is the larger so

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lets just say its negative of the three half minus two thirds which is equal to the negative if we use

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six as a common denominator if we use six as a common denominator three half well six is the denominator

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as nine six minus four over six that's what two third is equal to so nine minus four is five six let

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me have this negative value so equals negative five six so times p to the negative five six power times

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one half times that last one half here now three time three times three are twenty seven to the one third

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the cube root of twenty seven is three so this right here simplifies to three and then three to the one

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third that's cube root of three so we can say three and then this right here is the cube root of three

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the cube root of three so we get three times the cube of three time p to the negative five six to some

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time you wanna right p out here infront of the radical that's actually to be separate so iwill right

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the p to the negative five six p to the negative five six

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then i will right the radical so times the cube root of three and all that times one half so we can all

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of that time one half or you can just divide it by two and we have simplified it about as far as you

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can do as three half you can say three half is one point five p to the negative five six time the cube

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root of three if you wanted to you can calculate it cube root of three multiply that times one half you