Worked example: Cube root of a negative number

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.