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## Pre-algebra

### Course: Pre-algebra > Unit 11

Lesson 1: Square roots & cube roots- Intro to square roots
- Square roots of perfect squares
- Square roots
- Intro to cube roots
- Cube roots
- Worked example: Cube root of a negative number
- Equations with square roots & cube roots
- Square root of decimal
- Roots of decimals & fractions
- Equations with square roots: decimals & fractions
- Dimensions of a cube from its volume
- Square and cube challenge
- Square roots review
- Cube roots review

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# Dimensions of a cube from its volume

When we know the volume of a cube, we can use the cube root of the volume to find the length of each side. We'll need to factor the volume and find 3 equal groups of factors. The value of one group is the length of one side of the cube.

## Want to join the conversation?

- Is there an easier, simpler, way of explaining it?(5 votes)
- honestly, memorize your squares and cubes for 1-12, those are what you'll mostly see, you can use a calculator for higher up ones, this is probably the most simple way, but theres other ways explaining this, try looking on youtube(22 votes)

- What is 5 cubic ft to inches(0 votes)
- It does not make sense to convert from cubic ft to inches, because cubic ft is a unit of volume and inches is a unit of length.

So I'll assume you meant to ask "what is 5 cubic ft to cubic inches". There are 12^3 = 1,728 cubic inches in a cubic ft (a cubic ft can be visualized as a 12 by 12 by 12 array of cubic inches, so this is why there are 1,728 instead of just 12 cubic inches in a cubic ft).

Therefore, 5 cubic ft is 5*1,728 = 8,640 cubic inches.

Have a blessed, wonderful day!(26 votes)

- it's getting easier :)(10 votes)
- what is 3000 to the power of 1000000? the Calculator won't answer me(4 votes)
- So I looked it up (bc i have no idea) and it literally told me that the answer is
**INFINITY**(4 votes)

- Didn't Sal already use 512 in “Worked example: Cube root of a negative number” So wouldn’t we already know the answer?(5 votes)
- I believe so, I just saw that video before this one.(4 votes)

- What about prime numbers?(4 votes)
- I’m not entirely sure what you mean.

If the volume you’re given isn’t a perfect cube, you can use a calculator to find the cube root of the volume to find each side length. You could also factor and simplify to get an exact answer for the cube root.

Does that help? If not, I’ll try to fix my answer.(5 votes)

- Hi li.yue.is.cool I think that we are allowed to use a calculator, but if we really want to learn the concept of the equation then it's best not to use a calculator. But if you already know how to do it, then that's when I'll usually say to use a calculator, it also matters on what your teacher says.(5 votes)
- To do this for a polyhedron, would you use logorythms?Is it doable?I mean a polyhedron besides a cube.(7 votes)
- are we allowed to use calculator?(4 votes)
- in the answer do you write the cube root of 8^3 or just x=8?(3 votes)
- it wouldn't be 8^3 because khan academy doesn't understand that symbol, so it would be x=8(3 votes)

## 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.