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we're asked to find the cube root of negative 343 or another way to think about it is some number that when I multiply it by itself three times I'm going to get negative 343 or another way to view it this is the same thing as negative 343 to the 1/3 power and the best way to do this is to really just try to factor this out so the first thing we could do so let me just factor negative 343 so the first thing I'd like to do is just factor out the negative one so the same thing as negative one times 343 and let's think about this is this divisible by two no is it just by three see the digits do not add up to a multiple of three they add up to ten so it's not divisible by three not divisible by four since it's odd not just by five because it doesn't end with a five or zero it's not divisible by six because it's not divisible by 2 or three is it divisible by seven let's check this out so 7 goes into 343 seven goes into 34 4 times 4 times 7 is 28 4 34-28 is going to be 6 bring down the 3 7 goes into 63 exactly 9 times so and then we end up so 9 times 7 is 63 no remainder so this is going to be 7 times 49 and we know that 49 is the same thing as 7 times 7 so how can we rewrite this this is the same thing as taking is the same thing as taking the cube root the cube root of negative 1 of negative 1 times 7 times 7 times 7 which is the same thing as taking the cube root of negative 1 times the cube root the cube root of 7 times 7 times 7 now what's the cube root of negative 1 well negative 1 times itself 3 times is negative 1 so this right here is negative 1 you can verify negative 1 times negative 1 times negative 1 is indeed negative 1 this becomes positive 1 multiplied by negative 1 again you get negative so this is negative one and then this over here the cube root of 7 times 7 times 7 well that's just going to be 7 7 multiplied by itself 3 times gives us 7 times 7 times 7 or 343 so it's going to be negative 1 times 7 which is the same thing as negative 7 so our answer is negative 7 and we're done