Simplifying cube root expressions

multiply and simplify five times the cube root of 2x squared times three times the cube root of 4x to the fourth so the two things that pop out in my brain right here is that we can change the order a little bit because multiplication is both commutative well the commutative property allows us to switch the order for multiplication and so we can get the constant term so we can multiply the 5 times the 3 and then the other two things that we're multiplying they're both the cube root which is the same thing as taking something to the 1/3 power so the cube root of x this is exactly the same thing as raising X to the 1/3 so let's do that let's switch the order and let's rewrite these cube roots as raising it to the 1/3 power so I have the 5 and the 3 so that's going to be 5 times 3 and then we have the cube root of do that in a new color then we have the cube root of 2x squared so this I can rewrite as 2x squared to the 1/3 power and then I have the cube root of 4x to the 4th so that's the same thing as 4x to the 4th to the 1/3 power and now we know from our exponent properties if we have two things that are both raised to the same power and then we take the product we could just take the product first and then raise it to the power so if I have a a to the X power times B to the X power this is the same thing as a times B to the X power so we can simplify this part of the expression right over here as 2x squared 2x squared times 4x to the fourth to the 1/3 power and of course 5 times 3 is 15 and if we simplify what's in the expression right over here we once again it's commutative so we can swap the order and and it's associative and so we can swap the groupings or the grouping how we group them doesn't matter because it's all multiplication here this is 2 times 4 which is 6 times x squared times X to the fourth x squared times X to the fourth is X to the sixth power and it's all of that to the 1/3 power and then that is times that oh sorry not 6 2 times 4 is 8 what am i doing two times four is eight two times four is eight x squared times X 2x squared times X to the fourth is X to the sixth I think my brain was adding the exponents and wrote the six down of course two times four is eight not six but we add the exponents they have the same base x squared times x squared times X to the fourth is X to the sixth and we're going to raise that to the 1/3 power and then all of that is times 15 and then we essentially can use this property yet actually not that property we know that this we know if I have something well actually exactly this property again we have something multiplied to a power this is the exact same thing this is the exact same thing as 8 to the 1/3 power times X to the sixth to the 1/3 power and then all of that is being multiplied all of that is being multiplied by 15 and so 8 to the 1/3 power that's the same thing as the cube root of 8 you might recognize that 8 is 2 times 2 times 2 so 8 to the 1/3 power is 2 8 is 2 to the third so 2 to the third to the 1/3 is 2 to the first 2 times 2 times 2 is 8 and X to the 6 to the 1/3 we know from our exponent properties that's the same thing as X to the 6 times 1/3 power or X to the 6 divided by 3 power or 6 divided by 3 is 2 or x squared so that is just x squared see a 15 times 2 which gives us 30 so that's these terms right over here and then you have this term right over here I want to do that in a different color and then you have this term right over that's not a different color you have this term right over here is x squared and you are done it there's a bunch of ways you could do it you might not decide to use exponent notation you could say look if this is cube root this is a cube root I can then take the cube root of the product of both of them so you don't have to write it one third here you could just write the cube root of this whole thing over here and then depending on how you want to group it in all the rest you could do this in different orders as long as you get the right exponent properties you should be getting you should get to this same answer