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 what is 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 get it just 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 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 which is the same thing as the cube root of negative 1 times the cube root 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 one well I get negative one this right here is negative one negative 1 to the third power is equal to negative one times negative one times negative one which is equal to negative one so the cube root of negative 1 is negative one so it becomes negative one times times this business right here times the cube root cube root of 512 and let's think what this might be so let's 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 3 times 64 is 64 is 2 times 32 32 is 2 times 16 we're getting a lot of twos here 16 is 2 times 2 times 8 8 is 8 is 2 times 2 times 4 and 4 is 2 times 2 so we got a lot of tubes if you multiply so essentially you multiply two one two three four five six seven eight nine times you're going to get 512 or two 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 it three groups of twos or can we if we can also find let me look at that look at it this way we can find it three groups of two twos over here so that's two times two is four two times two is four 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 this is equal to negative 1 or I could just put a negative sign here negative 1 times the cube root the cube root of 8 times 8 times 8 so we're asking our question what number can be multiplied by itself three times or ticket 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 is negative 8 and we are done and you can verify this multiply negative 8 times itself 3 times well let's just do it negative 8 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