Rewriting mixed radical and exponential expressions

we're asked to simplify our to the two-thirds s to the third that whole thing squared times the square root of 20 R to the fourth s to the fifth this looks kind of daunting but I think if we take it step by step it shouldn't be too bad so first we can look at this first expression right here where we're taking this product to the second power we know that instead we can take each of these each of the terms in the product to the second power and then take the product so this is going to be the same thing as R to the 2/3 squared times s to the third squared and now let's look at this radical over here we have the square root but that's the exact same thing as raising something to the one-half power so this is equal to so times this part let me do this in a different color this part right here that is the same thing as 20 and instead of just writing 20 let me write 20 is the product of a perfect square in a non perfect square so 20 is the same thing as 4 times 5 that's the 20 part times R to the 4th times s to the fifth now let me write s to the 5th also as a product of a perfect square and a non-perfect square R to the 4th is obviously a perfect square it's square root of R squared let's write s to the fifth in a similar way so s to the fifth we can rewrite as s to the fourth times s right s to the fourth times s to the first that is s to the fifth and of course all of this has to be raised to the one-half power now let's simplify this even more if we're taking something to the 2/3 power and then to the 2nd power we can just multiply the exponents so this term right here we can simplify this as R to the 4/3 power and just as a bit of review taking something to the 4/3 power you can view it as either taking finding its cube root taking it to the 1/3 power and then taking its cube root to the 4th power or you can view it as taking it to the 4th power and then finding the cube root of that those are both legitimate ways of sum being something being raised to the 4/3 power so you have R to the 4/3 times s to the 3 times 2 times s to the 6th power or s to the sixth power and then we could raise each of these terms right here to the one-half power so times let me color code it a little bit times and we actually wouldn't need the parentheses once we do that times 4 to the 1/2 times 4 to the 1/2 times times 5 to the 1/2 that's that term right there times R to the fourth to the 1/2 R to the fourth to the 1/2 power times might run out of colors s to the fourth to the 1/2 power s to the fourth to the 1/2 power raising each of these terms to that 1/2 power x times s to the 1/2 power and there's a lot of ways we can go with this but the one thing that might jump out is that there are some perfect squares here and we're raising them to the 1/2 power we're taking their square root so let's simplify those so this 4 to the 1/2 that's the same thing as 2 we're taking the principal root of 4 five to the one-half well we can't take the square root of that so let's just write that as the square root of five square root of five R to the fourth to the 1/2 4 times 1/2 there's two ways you can think about it 4 times 1/2 is 2 so this is R squared or you could say the square root of R to the fourth is R squared so this is R squared similarly s the square root the square root of s to the fourth or s to the fourth to the one-half is also s squared and then this s to the one-half let's just write that as the square root of s square root of s just like that now let's see what else we can do here so we have we have let me write these other terms we have an R to the 4/3 times s to the sixth times 2 times square root of 5 times R squared times s squared times the square root of s now a couple of things we can do here we could combine these s terms let's do that I should just write the 2 out front first so let's write the 2 out front first so you have 2 times now let's look at these 2's terms over here we have s to the sixth times s square and when someone says to simplify it there's multiple interpretations for it but we'll just say s to the 6 times s squared that's s to the 8th right 6 plus 2 times s to the 8th power times now this one's interesting and we might want to break it up depending on what we consider something to what we consider to be truly simplified we have R to the 4/3 times R squared R to the 4/3 times R squared R to the 4/3 is the same thing as R to the 1 and 1/3 that's what 4/3 is so 1 and 1/3 1 and 1/3 plus 2 is 3 and 1 thirds so we could write this times R to the 3 and 1/3 and it's a little consistent over here I'm adding a fraction over here with the s I kind of left out the s to the 1/2 from the S is here but we could play around with it and all of those would be valid expression so we've already dealt with the 2 we've already dealt with these two S's we've already dealt with these RS and then you have the square root of 5 times the square root of s and we could merge them if we want but I won't do it just yet times the square root of 5 times the square root times the square root of s now there's two ways we could do it we might not like having a fractional exponent here and then we could break it out or we might want to take this guy and merge it with the eighth power because you know that this is the same thing as s to the one-half so let's do it both ways so if we wanted to if we wanted to merge all of the exponents we could write this as 2 times s to the 8th times s to the 1/2 so s to the 8th and s to the 1/2 that would be 2 times s to the 8th I can even write as a decimal 8.5 right 8 plus this you could imagine this is s to the 0.5 power so that's 8 point 5 times R to the 3 and 1/3 I'm kind of mixing notations here that I have just a decimal notation then I have a fraction notation mixed number notation times the square root of 5 times the square root of 5 this is one simplification I kind of have it in the fewest terms possible the other simplification if you don't want to have these fractional exponents out here you could write it as I'll do this in a different color you could write this and these are all equivalent statements so it's up to debate what simplified really means so you could write this as 2 times s to the 8th s to the 8th instead of writing R to the 3 and 1/3 we could write R to the third R to the third times the cube root of R which is the same thing as R to the 1/3 we could write R to the third times R to the 1/3 our to the 1/3 is the same thing is the cube root of R and then you have the square root of these two guys both of these guys are being raised to the one-half power so you could then say times the square root of 5s I like this one a little bit more the one on the left to me this is really simplified we've merged all of the bases you know we are we have these two numbers here we've merged all the s terms all the are terms this is a little bit more complicated you have a cube root you have separated the S is in the RS so I would go with this one if someone really wanted me said hey Sal simplify it how you like