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We are asked to find the
cube root of negative 512. Or another way to think about
it is if I have some number, and it is equal to the
cube root of negative 512, this just means that
if I take that number and I raise it to the third
power, then I get negative 512. And if it 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 negative 512, the best thing to do is to just
do a prime factorization of it. And before we do a prime
factorization of it to see which of these factors
show up at least three times, let's at least think about the
negative part a little bit. So negative 512, that's
the same thing-- so let me rewrite 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 of negative 1 times the cube root of 512. And this one's pretty
straightforward to answer. What number, when I raise
it to the third power, do I get negative 1? Well, I get negative 1. This right here is negative 1. Negative 1 to the third power
is equal to negative 1 times negative 1 times negative 1,
which is equal to negative 1. So the cube root of
negative 1 is negative 1. So it becomes negative 1
times this business right here, times the
cube root of 512. And let's think
what this might be. So let's 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. So essentially, if
you multiply 2 one, two, three, four, five, six,
seven, eight, nine times, you're going to get 512, or
2 to the ninth 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's
definitely three twos here. But can we find
three groups of twos, or we could also find--
let me look at it this way. We can find three groups
of two twos over here. So that's 2 times 2 is 4. 2 times 2 is 4. So definitely 4 multiplied
by itself three 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. This is 2 times 2 times 2. That's 8. 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 rewrite
this expression right over here as the cube
root of 8 times 8 times 8. So this is equal to
negative 1, or I could just put a negative sign
here, negative 1 times the cube root of
8 times 8 times 8. So we're asking our question. What number can we multiply
by itself three 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 simplify to 8. And so our answer to this,
the cube root of negative 512, is negative 8. And we are done. And you could verify this. Multiply negative 8
times itself three times. Well, let's just do it. Negative 8 times negative
8 times negative 8. Negative 8 times negative
8 is positive 64. You multiply that times negative
8, you get negative 512.