Simplifying a cube root

We're are 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 that 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 1. So this is the same thing as negative 1 times 343. And let's think about this. Is this divisible by 2? No. Is it divisible by 3? Let's see-- the digits do not add up to a multiple of 3. They add up to 10. So it's not divisible by 3. Not divisible by 4 since it's odd. Not divisible by 5 because it doesn't end with a 5 or a 0. It's not divisible by 6, because it's not divisible by 2 or 3. Is it divisible by 7? Let's check this out. So 7 goes into 343-- 7 goes into 34 four times. 4 times 7 is 28. 34 minus 28 is going to be 6. Bring down the 3. 7 goes into 63 exactly nine times. So 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 the cube root 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 of 7 times 7 times 7. Now, what's the cube root of negative 1? Well, negative 1 times itself three times is negative 1. So this right here is negative 1. You could verify it. Negative 1 times negative 1 times negative 1 is indeed negative 1. This becomes positive 1. Multiply by negative 1 again, you get negative 1. So this is negative 1. 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 three 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.