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multiply and simplify five times the cube root of two x squared

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times three times the cube root of four x to the fourth

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so the two things that pop out of my brain right here is that we can change

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the order a little bit cause multiplication is both commutative

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well the commutative property allows us to switch the order for multiplication

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so we can get the constant terms we can multiply the five times the three

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and then the other two things that we are multiplying They are both the cube root which is the

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same thing as taking something to the one-third power

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so the cube root of x. This is exactly the same thing as raising x to the one-third

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So let's do that. Let's switch the order and let's rewrite these cube roots as

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raising it to the one-third power

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so I have the five and the three so that's going to be five times three

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and then we have the cube root of

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I will do that in a new color

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then we have the cube root of two x squared

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so this I can rewrite as two x squared to the one-third power

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,and then I have the cube root of four x to the fourth

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so that's the same thing as four x to the fourth to the one-third power

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and now we know from our exponent properties: if we have two things that are both

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raised to the same power and then we take the product

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we could just take the product first and then

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raise it to the power

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so if I have a to the x power times b to the x power

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This is the same thing as a times b to the x power

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So we can simplify this part of the expression right over here

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as two x squared

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time four x to the fourth to the

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one-third power and of course five times three is fifteen

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and if we simplify what is in the expression right over here we

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once again it is commutative so we can swap the order and

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it is associative so we can

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swap the grouping or how we group them does not matter

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cause it is all multiplication here

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This is two times four which is six times x squared times x to the fourth

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x squared times x to the fourth

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is x to the sixth power

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and it is all of that to the one-third power

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and then that is times

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oh sorry not six two times four is eight what am I doing?

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two times four is eight

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x squared times x to the fourth is

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x to the sixth

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I think my brain was adding the exponents and wrote the six down

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of course two times four is eight not six but we add the exponents

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they had the same base

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x squared times x to the fourth is x to the sixth

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and we are gonna raise that to the one-third power

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and then all of that is times fifteen

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and then we essentially can use this property again

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actually not that property

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We know that this. We know if I have something

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well actually exactly this property again

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We have something multiplied to a power

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This is the exact same thing

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as eight to the one-third power times

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x to the sixth to the one-third power

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and then all of that is being multiplied

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by fifteen and so eight to the one-third power

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this is the same thing as the cube root of eight

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you might recognize that eight is two times two times two

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so eight to the one-third power is two

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eight is two to the third so two to the third

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to the one-third is two to the first

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two times two times two is eight

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and x to the sixth to the one-third

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we know from our exponent properties that is the same thing as

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x to the six times one-third power

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x to the six divided by three power

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or six divided by three is two or x squared so that is just x squared

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so we have

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fifteen times two which gives us thirty

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so that is these terms right over here

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and then you have this term right over here

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I want to do that in a different color

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and then you have this term right over here

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that's not a different color! You have this term

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right over here is x squared

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and you are done

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and there is a bunch of ways you could do it

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You might not decide to use exponent notation

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You could say look this is a cube root

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This is a cube root I can take the cube root of the product of both of them

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so you do not have to write the one-third here

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you could just write the cube root of this whole thing over here

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and then depending on how you want to group it and all the rest

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you could do this in different orders

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as long as you get the right exponent properties

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you should be getting

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you should get to this same answer