We are asked to find the cube root of -512
Or another way to think about it is if I have some number
and it is equal to the cube root of -512
This just means if I take that number and I raise it to the to the third power
then I get -512. And if doesn't jump out at you immediately
what this is the cube of or what we have to raise to the third power to get -512
the best thing to do, is just to do a prime factorization of it.
And before we do a prime factorization of it and to see which of these factors show up at least 3 times,
lets at least think about the negative part a little bit.
So negative 512 thats the same thing, so let me re-write the expression
this is the same thing as the cube root of negative 1 times 512.
Which is the same thing as the cube root,
which is the same thing as the cube root of -1 times the cube root of 512
and this one is pretty straight forward to answer.
What number, when I raise it to the third power, do I get -1?
Well I get -1! This right here is -1
-1 to the third power is equal to -1 time-1 times -1, which is equal to -1
So the cube root of -1 is -1
So it becomes -1 times, times this business right here
Times the cube root, cube root of 512 and lets think of what this might be
So, so lets do the prime factorization
So 512 is 2 times 256
256 is 2 times 128
128 is 2 times 64
We already see a 2 three times
64 is 2 times 32
32 is 2 times 16
We're getting a lot of twos here
16 is 2 times 8
8 is 2 times 4
And 4 is 2 times 2
So we got a lot of twos
If you multiply
So essentially if you multiply two 1, 2, 3, 4, 5, 6, 7, 8, 9 times
You are going to get 512
Or 2 to the 9th power is 512
And that by itself should give you a clue of what the cube root is
But another way to think about it is
Can we find
There is definetly 3 twos here
But can we find 3 groups of 2
Or can we
We can also find
Look at it this way
We can find three groups of two 2s over here
So that's 2 times 2 is 4
2 times 2 is 4
So definitely 4 multiplied by itself 3 times is divisible into this
But even better it looks like we can get three groups of three twos
So one group, two groups and three groups
So each of these groups
2 times 2 times 2, that's 8
That is 8
And this is 2 times 2 times 2, thats eight
And this is also 2 times 2 times 2
So that's 8
So we could write 512 as being equal to
8 times 8 times 8
And so we can re-write this expression right over here
As the cube root of 8 times 8 times 8
And so this is equal to
this is equal to -1
Or I'll just put a negitave sign here
-1 times the cube root, the cube root of 8 times 8 times 8
So we are asking the question
What number can we multiply by itself 3 times
Or to the third power to get 512
Which is the same thing as 8 times 8 times 8
Well clearly this is 8
So the answer
This part right over here is just going to simplifly to eight
And so our answer to this
The cube root of -512 is -8
And we are done
And you can verify this
Multiply -8 times itself 3 times
Well lets just do it
-8 times -8 times -8
-8 times -8 is positive 64
You multiply that times negative 8
You get -512