Main content

## 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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# Worked example: Cube root of a negative number

Learn how to find the cube root of -512. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- i still do not get it :( please explain more, for example where did he get the 2's?(15 votes)
- I find it easiest just to Guess and Check....What number cubed equals -512? I find people start to memorize the common ones after a bit.(8 votes)

- How do i get the cube root of a number like 4?(8 votes)
- The cube root won't be an exact number, if that's what you're asking.

To get an idea of what it might be,

4 is found in between 1 and 8

This means that the cube root of 4 will be in between cube root of 1 and cube root of 8, or in other words, between 1 and 2.

4 is closer to 1 than 8 (4-1=3 while 8-4=4). so the cube root of 4 will be a number in between 1 and 1.5(16 votes)

- the cube root does not have to be perfect?(7 votes)
- That is true just like square roots. Any whole number that is not 1, 8, 27, 64, 125 ... is not a perfect cube.(6 votes)

- How to simplify the cube root of 9317 ... any ideas?(4 votes)
- The simplified cube root of that number is 11 3√ 7(7 votes)

- Hang on a second... at1:10, how is the cube root of -1 = -1? Wouldn't it be -i?(5 votes)
- No because -1 x -1 x-1 = -1. When three negatives are multiplied together 2 of them cancel out, but on is left so no there is no change to the sign.(5 votes)

- Can someone please explane this to me this is hard(4 votes)
- First you do the prime factorization then you make three identical groups of the final numbers. Finally you multiply the numbers together in those groups. This might not help it is my first reply. :((3 votes)

- At1:55, couldn't Sal have stop factoring at 64 since 8x8=64?(4 votes)
- No, because 8 isn't a prime number. when doing prime factorization, you have to factor until everything is prime.(3 votes)

- 2x3=6 not 8 rite?(3 votes)
- 2 x 3 can either be thought of as 2 + 2 + 2 or as 3 + 3. Or stating that another way, 2 groups of 3 items or 3 groups that each have 2 items and it does equal 6.

To cube something is to multiply it by itself, 2 x 2 x 2 = 4 x 2 = 8. It is written 2^3 = 8(5 votes)

- wait a minute...

Didn't Sal say before this that the number being simplified cant be negative?

like when he says that Xcubed= (-512), how can 512 be negative?(2 votes)- 512 can be negative because 3 numbers that are negative can be multiplied to make a negative number. For example:

(-2)*(-2)*(-2)= -8

Because (-2)*(-2) is 4, and then if you multiply it by -2 again, then it is -8, which is a negative number.(5 votes)

- What is the square root of a negative number?(3 votes)
- Zero has one square root which is 0. Negative numbers don't have real square roots since a square is either positive or 0. The square roots of numbers that are not a perfect square are members of the irrational numbers. This means that they can't be written as the quotient of two integers.(2 votes)

## Video transcript

We are asked to find the
cube root of negative 512. Or another way to think about
it is if I have some number, and it is equal to the
cube root of negative 512, this just means that
if I take that number and I raise it to the third
power, then I get negative 512. And if it doesn't jump out
at you immediately what this is the cube
of, or what we have to raise to the third
power to get negative 512, the best thing to do is to just
do a prime factorization of it. And before we do a prime
factorization of it to see which of these factors
show up at least three times, let's at least think about the
negative part a little bit. So negative 512, that's
the same thing-- so let me rewrite the
expression-- this is the same thing as the cube
root of negative 1 times 512, which is the same thing as
the cube root of negative 1 times the cube root of 512. And this one's pretty
straightforward to answer. What number, when I raise
it to the third power, do I get negative 1? Well, I get negative 1. This right here is negative 1. Negative 1 to the third power
is equal to negative 1 times negative 1 times negative 1,
which is equal to negative 1. So the cube root of
negative 1 is negative 1. So it becomes negative 1
times this business right here, times the
cube root of 512. And let's think
what this might be. So let's do the
prime factorization. So 512 is 2 times 256. 256 is 2 times 128. 128 is 2 times 64. We already see a 2 three times. 64 is 2 times 32. 32 is 2 times 16. We're getting a
lot of twos here. 16 is 2 times 8. 8 is 2 times 4. And 4 is 2 times 2. So we got a lot of twos. So essentially, if
you multiply 2 one, two, three, four, five, six,
seven, eight, nine times, you're going to get 512, or
2 to the ninth power is 512. And that by itself
should give you a clue of what the cube root is. But another way
to think about it is, can we find-- there's
definitely three twos here. But can we find
three groups of twos, or we could also find--
let me look at it this way. We can find three groups
of two twos over here. So that's 2 times 2 is 4. 2 times 2 is 4. So definitely 4 multiplied
by itself three times is divisible into this. But even better,
it looks like we can get three groups
of three twos. So one group, two
groups, and three groups. So each of these groups, 2
times 2 times 2, that's 8. That is 8. This is 2 times 2 times 2. That's 8. And this is also
2 times 2 times 2. So that's 8. So we could write 512 as being
equal to 8 times 8 times 8. And so we can rewrite
this expression right over here as the cube
root of 8 times 8 times 8. So this is equal to
negative 1, or I could just put a negative sign
here, negative 1 times the cube root of
8 times 8 times 8. So we're asking our question. What number can we multiply
by itself three times, or to the third
power, to get 512, which is the same thing
as 8 times 8 times 8? Well, clearly this is 8. So the answer, this
part right over here, is just going to simplify to 8. And so our answer to this,
the cube root of negative 512, is negative 8. And we are done. And you could verify this. Multiply negative 8
times itself three times. Well, let's just do it. Negative 8 times negative
8 times negative 8. Negative 8 times negative
8 is positive 64. You multiply that times negative
8, you get negative 512.