Dimensions of a cube from its volume

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Video transcript

- [Voiceover] Let's say that we had a cube, let me draw the cube here, So we have a cube, and we know that the volume of this cube is equal to 512 cubic centimeters. So my question to you is, what are the dimensions of this cube? So what is this length gonna be? What is this, I guess you could say depth, and what is this height going to be? And, we know it's a cube, so these are all going to be equal. And so like always, encourage you to pause the video and try to figure it out. Well let's call this length x. If that's x then this is going to be x, and then this is x as well. So if the volume is 512 cubic centimeters, that means that x times x times x is going to be equal to 512. Is going to be equal to 512, or we could say that x to the third power is equal to 512, or we could say that x is equal to the cube root of 512. So what's the cube root of 512? And the easiest way I can think about doing this, if I don't have a calculator, is to just try to do a prime factorization of this by hand. So that's what I'm going to attempt to do. So let's see, does two go into 512? Sure, 512s even, so this is going to be two times, let's see, 256, yeah, two times 256. 256 that's also divisible by two. That's two times 128, which is also divisible by two, that's two time 64, which is also divisible by two. That's two times 32, let's see, I can keep going, that's two times 16. Which is two times eight, which is two times four, which is two times two. So 512 that's the same thing as two to the, let's see you have, one, two, three, four, five, six, seven, eight, nine. That's two to the ninth power. But what we care about is what times itself times itself is equal, what number, if I have three of 'em and I multiply 'em together, get us to 512? And to think about that, we could say, "Look, I have nine numbers here. "So let me divide into three groups." So if this is one group, and this is the next group, and then this is the next group right over here, we could say that 512 is the same thing as two times two times two, which is eight, times two times two times two, which is eight, times two times two times two. So 512 is the same thing as eight to the third power. So we could say, that, x, I'll do it over here, x is equal to the cube root of, instead of writing 512, instead of writing 512, I could write eight to the third power. Now, what's the cube root of something to the third power? Well, it's just gonna be this something. So x, x is going to be equal to eight. So if the volume here is 512 cubic centimeters, each dimension is going to be eight centimeters. So x is equal to eight centimeters. This is equal to eight centimeters. I'm just writing the units now. This is equal to eight centimeters, and we're done. But, if you didn't know offhand that eight to the third power is 512, this is a reasonable way of coming to that conclusion. Anyway, hopefully that helped.