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We are asked to find the cube root of -512

2
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Or another way to think about it is if I have some number

3
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and it is equal to the cube root of -512

4
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This just means if I take that number and I raise it to the to the third power

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then I get -512. And if doesn't jump out at you immediately

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what this is the cube of or what we have to raise to the third power to get -512

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the best thing to do, is just to do a prime factorization of it.

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And before we do a prime factorization of it and to see which of these factors show up at least 3 times,

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lets at least think about the negative part a little bit.

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So negative 512 thats the same thing, so let me re-write the expression

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this is the same thing as the cube root of negative 1 times 512.

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Which is the same thing as the cube root,

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which is the same thing as the cube root of -1 times the cube root of 512

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and this one is pretty straight forward to answer.

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What number, when I raise it to the third power, do I get -1?

16
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Well I get -1! This right here is -1

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-1 to the third power is equal to -1 time-1 times -1, which is equal to -1

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So the cube root of -1 is -1

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So it becomes -1 times, times this business right here

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Times the cube root, cube root of 512 and lets think of what this might be

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So, so lets do the prime factorization

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So 512 is 2 times 256

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256 is 2 times 128

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128 is 2 times 64

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We already see a 2 three times

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64 is 2 times 32

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32 is 2 times 16

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We're getting a lot of twos here

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16 is 2 times 8

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8 is 2 times 4

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And 4 is 2 times 2

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So we got a lot of twos

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If you multiply

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So essentially if you multiply two 1, 2, 3, 4, 5, 6, 7, 8, 9 times

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You are going to get 512

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Or 2 to the 9th power is 512

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And that by itself should give you a clue of what the cube root is

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But another way to think about it is

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Can we find

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There is definetly 3 twos here

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But can we find 3 groups of 2

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Or can we

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We can also find

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Look at it this way

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We can find three groups of two 2s over here

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So that's 2 times 2 is 4

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2 times 2 is 4

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So definitely 4 multiplied by itself 3 times is divisible into this

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But even better it looks like we can get three groups of three twos

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So one group, two groups and three groups

51
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So each of these groups

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2 times 2 times 2, that's 8

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That is 8

54
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And this is 2 times 2 times 2, thats eight

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And this is also 2 times 2 times 2

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So that's 8

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So we could write 512 as being equal to

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8 times 8 times 8

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And so we can re-write this expression right over here

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As the cube root of 8 times 8 times 8

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And so this is equal to

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this is equal to -1

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Or I'll just put a negitave sign here

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-1 times the cube root, the cube root of 8 times 8 times 8

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So we are asking the question

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What number can we multiply by itself 3 times

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Or to the third power to get 512

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Which is the same thing as 8 times 8 times 8

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Well clearly this is 8

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So the answer

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This part right over here is just going to simplifly to eight

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And so our answer to this

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The cube root of -512 is -8

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And we are done

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And you can verify this

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Multiply -8 times itself 3 times

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Well lets just do it

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-8 times -8 times -8

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-8 times -8 is positive 64

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You multiply that times negative 8

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You get -512