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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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# Intro to cube roots

CCSS.Math:

Learn the meaning of cube roots and how to find them. Also learn how to find the cube root of a negative number.

## Want to join the conversation?

- Is there any easy method for finding a square root .... especially for bigger numbers like 1225 or etc..(24 votes)
- Use a calculator. Easily the easiest way to find any kind of root, without wasting time trying to do it by hand. If you want to find out how to do it manually though, try this. https://www.homeschoolmath.net/teaching/square-root-algorithm.php(33 votes)

- Is there any easy way to memorize cube roots? Is there a pattern or some way to help mentally solve cube roots without using a calculator or doing any work?

Thanks(11 votes)- If the last digit of a cube root is 2 then the unit digit will be 8. If the last digit of a cube root is 3 then the unit digit will be 7. If the last digit of a cube root is 7 then the unit digit will be 3. If the last digit of a cube root is other than 2, 3, 7 and 8 then put the same number as the unit digit.(13 votes)

- What are imaginary numbers? i forget(8 votes)
- An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i² = −1. The square of an imaginary number bi is −b². For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.(20 votes)

- can a cube root be a negative?(5 votes)
- Yes this is possible. The cube root of a negative number is negative. For example, the cube root of -8 is -2, because (-2)^3 = (-2)(-2)(-2) = -8.

Have a blessed, wonderful day!(24 votes)

- Sal sounds like my doctor.(11 votes)
- If you are reading this I hope you have a wonderful day and life treats you well, no matter what the situation remember to stay positive and push through cause your a strong person and you'll have good luck soon:)(9 votes)
- Thank you so much! I really needed this today! You're right, life is full of challenges and belly jelly😅 🤬 Please post more encouragement sometimes because it really helps my inner being.(6 votes)

- I'm still really confused about cube roots and all. I went through some of the explanations but they used all of these fancy words or numbers and I wish there was an explanation for this which explained it as simply as possible.(6 votes)
- When you cube a number, you raise it to an exponent of 3.

For example: 2^3 = 2*2*2 = 8

A cube root reverses this process. You are being asked to find the number that was originally "cubed".

For example: cube root(8) = 2 because 2^3 = 8

Hope this helps.(11 votes)

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⡄⠀⣿⢹⡇⢸⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿(9 votes) - whoever see this. have a good day(9 votes)
- So, you multiply kind of like finding a shapes volume?(6 votes)
- Yes, what you said, but in reverse, is the geometric meaning of taking a cube root. The cube root of a number can be thought of as the edge length of a cube whose volume is that number.

Have a blessed, wonderful day!(5 votes)

## Video transcript

- [Voiceover] We already know a little bit about square roots. For example, if I were to tell you that seven squared is equal to 49, that's equivalent to saying that seven is equal to the square root of 49. The square root essentially unwinds taking the square of something. In fact, we could write it like this. We could write the square root of 49, so this is whatever number times itself is equal to 49. If I multiply that number times itself, if I square it, well I'm going to get 49. And that's going to be
true for any number, not just 49. If I write the square root of X and if I were to square it, that's going to be equal to X and that's going to be true for any X for which we can evaluate the square root, evaluate the principle root. Now typically and as you advance in math you're going to see that this will change, but typically you say,
okay if I'm going to take the square root of something,
X has to be non-negative. X has to be non-negative. This is going to change
once we start thinking about imaginary and complex numbers, but typically for the
principle square root, we assume that whatever's
under the radical, whatever's under here, is
going to be non-negative because it's hard to square a number at least the numbers that we know about, it's hard to square them
and get a negative number. So for this thing to be
defined, for it to make sense, it's typical to say
that, okay we need to put a non-negative number in here. But anyway, the focus of this video is not on the square root, it's
really just to review things so we can start thinking
about the cube root. And as you can imagine,
where does the whole notion of taking a square of something
or a square root come from? Well it comes from the notion of finding the area of a square. If I have a square like this
and if this side is seven, well if it's a square, all the
sides are going to be seven. And if I wanted to find the area of this, it would be seven times
seven or seven squared. That would be the area of this. Or if I were to say, well what is if I have a square, if I have, and that doesn't look
like a perfect square, but you get the idea, all the
sides are the same length. If I have a square with area X. If the area here is X, what are the lengths of
the sides going to be? Well it's going to be square root of X. All of the sides are
going to be the square root of X, so it's going to be the square root
of X by the square root of X and this side is going to
be the square root of X as well and that's going to be
the square root of X as well. So that's where the term
square root comes from, where the square comes from. Now what do you think cube root? Well same idea. If I have a cube. If I have a cube. Let me do my best attempt at
drawing a cube really fast. If I have a cube and a
cube, all of it's dimensions have the same length so this
is a two, by two, by two cube, what's the volume over here. Well the volume is going to
be two, times two, times two, which is two to the
third power or two cubed. This is two cubed. That's why they use the word cubed because this would be the volume of a cube where each of its sides have length two and this of course is
going to be equal to eight. But what if we went the other way around? What if we started with the cube? What if we started with this volume? What if we started with a cube's volume and let's say the volume
here is eight cubic units, so volume is equal to eight and we wanted to find
the lengths of the sides. So we wanted to figure out what X is cause that's X, that's X, and that's X. It's a cube so all the
dimensions have the same length. Well there's two ways that
we could express this. We could say that X times X times X or X to the third power is equal to eight or we could use the cube root symbol, which is a radical with a
little three in the right place. Or we could write that X is equal to, it's going to look very
similar to the square root. This would be the square root of eight, but to make it clear,
they were talking about the cube root of eight, we would write a little three over there. In theory for square root, you could put a little two over here, but that'd be redundant. If there's no number here, people just assume that
it's the square root. But if you're figuring out the cube root or sometimes you say the third root, well then you have to say, well you have to put this
little three right over here in this little notch in the
radical symbol right over here. And so this is saying X
is going to be some number that if I cube it, I get eight. So with that out of the way, let's do some examples. Let's say that I have... Let's say that I want to calculate the cube root of 27. What's that going to be? Well if say that this is
going to be equal to X, this is equivalent to
saying that X to the third or that 27 is equal to
X to the third power. So what is X going to be? Well X times X times X is equal to 27, well the number I can think of is three, so we would say that X, let
me scroll down a little bit, X is equal to three. Now let me ask you a question. Can we write something like... Can we pick a new color? The cube root of, let
me write negative 64. I already talked about
that if we're talking the square root, it's fairly typical that hey you put a negative number in there at least until we learn
about imaginary numbers, we don't know what to do with it. But can we do something with this? Well if I cube something,
can I get a negative number? Sure. So if I say this is equal to X, this is the same thing as
saying that negative 64 is equal to X to the third power. Well what could X be? Well what happens if
you take negative four times negative four times negative four? Negative four times four is positive 16, but then times negative
four is negative 64 is equal to negative 64. So what could X be here? Well X could be equal to negative four. X could be equal to negative four. So based on the math that we know so far you actually can take the cube
root of a negative number. And just so you know, you
don't have to stop there. You could take a fourth root and in this case you'd have a four here, a fifth root, a sixth root, a seventh root of numbers and we'll talk about that later in your mathematical career. But most of what you're going to see is actually going to be square root and every now and then you're
going to see a cube root. Now you might be saying, well hey look, you know, you just knew that
three to the third power is 27, you took the cube root, you get X, is there any simple way to do this? And like you know if i give
you an arbitrary number. If I were to just say, I don't know, if I were to say cube root of 125. And the simple answer is, well the easiest way to
actually figure this out is actually just to do a factorization and particular prime
factorization of this thing right over here and then
you would figure it out. So you would say, okay
well 125 is five times 25, which is five times five. Alright, so this is the same
thing as the cube root of five to the third power, which of course, is going to be equal to five. If you have a much larger number here, yes, there's no very simple way to compute what a cube root or a fourth root or a fifth root might
be and even square root can get quite difficult. There's no very simple
way to just calculate it the way that you might
multiply things or divide it.