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Simplify the expression using rational exponents.

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So we have the cube root of all of this stuff over here which would be the same thing, or which is the same thing by definition as

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all of this stuff 64 a to the 6th, b to the 3rd, c to the 9th, raised to the 1/3rd power. These are equivalent, by definition.

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And if we take the product of a bunch of things and raise it to an exponent, thats the same thing as raising each of these things

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that we take the product of to the exponent first and then multiplying. So this is going to be equal to 64 to the 1/3rd power times

6
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a to the 6th to the 1/3rd power times b to the 3rd to the 1/3rd power times c to the 9th to the 1/3rd power. And we've seen this

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in several examples already. Well, first lets try to simplify this. 64 to the 1/3rd - you might recognize that already - but we can factor

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64 as , well, I won't do pure prime factorization, cause you'll see its just 4 to the 3rd power. Actually you could verify that for yourself,

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or I encourage you, if thats not obvious to see that this is 4 to the 3rd power - 64, multiply it out for yourself, or if you're

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like how did you know that, do the prime factorization. What you're going to do is, well, let me just show you. If you do the prime

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factorization, you have 2 times 32, 32 is 2 times 16, 16 is 2 times 8, 8 is 2 times 4, 4 is 2 times 2, so you can have 2 multiplied

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by itself 3 times, or you could have a ..oh! a 2 multiplied by itself 3 times, and that happens twice, or you could have 4 times

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4 times 4. So if you,you,it it didn't immediately jump into your head that 64 is 4 to the 3rd power, you can literally do a brute force prime

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factorization here and see that you have 6. You can factor this into 2 to the 6th power or, the same thing as 4 to the 3rd power. So I'll

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, I'll let you think about that a little bit. But this right here simplifies - 64 is 4 to the 3rd power, and you raise that to the

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1/3rd power, that is the same thing as - actually let me write it that way, then I can make it more interesting - so this right here,

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this is 4 to the 3rd power, so if I have 4 to the 3rd power, and that raised to the 1/3rd power, so if I want to rewrite that term to

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right over here, thats going to be 4 to the 3 thirds power - I'm literally multiplying the 3 times 1/3rd. 4 to the 3rd, and then

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that to the 1/3rd. Then over here, I have a to the 6th, and then that to the 1/3rd power. That is going to be a to the 6 times 1/3rd

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power, or a to the 6 thirds power. Then I have b to the 3rd, and then that raised to the 1/3rd, so thats going to be, we literally

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can multiply exponents here, so we have b to the 3 over 3 power, and then finally, we have c to the ninth, and then that raised, and then that raised to

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the 1/3rd power. So that is c to the 9 over 3 power. And if you simplify it, this first term here - 4 to the 3 thirds, thats just

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4 to the 1st power, so thats just 4. We have a to the 6 thirds - 6 thirds is just 2 - so that just becomes a squared. b to the 3

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thirds power - that's just b to the first , or b, and then finally c to the 9 thirds - that's just c to the third power. So let me

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write c to the 3rd here. And we are done.