Finding Cube Roots

u16_l1_t2_we2 Finding Cube Roots

We are asked to find the cube root of -512

Or another way to think about it is if I have some number

and it is equal to the cube root of -512

This just means if I take that number and I raise it to the to the third power

then I get -512. And if 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 -512

the best thing to do, is just to do a prime factorization of it.

And before we do a prime factorization of it and to see which of these factors show up at least 3 times,

lets at least think about the negative part a little bit.

So negative 512 thats the same thing, so let me re-write 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,

which is the same thing as the cube root of -1 times the cube root of 512

and this one is pretty straight forward to answer.

What number, when I raise it to the third power, do I get -1?

Well I get -1! This right here is -1

-1 to the third power is equal to -1 time-1 times -1, which is equal to -1

So the cube root of -1 is -1

So it becomes -1 times, times this business right here

Times the cube root, cube root of 512 and lets think of what this might be

So, so lets 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

If you multiply

So essentially if you multiply two 1, 2, 3, 4, 5, 6, 7, 8, 9 times

You are going to get 512

Or 2 to the 9th 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 is definetly 3 twos here

But can we find 3 groups of 2

Or can we

We can also find

Look at it this way

We can find three groups of two 2s over here

So that's 2 times 2 is 4

2 times 2 is 4

So definitely 4 multiplied by itself 3 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

And this is 2 times 2 times 2, thats eight

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 re-write this expression right over here

As the cube root of 8 times 8 times 8

And so this is equal to

this is equal to -1

Or I'll just put a negitave sign here

-1 times the cube root, the cube root of 8 times 8 times 8

So we are asking the question

What number can we multiply by itself 3 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 simplifly to eight

And so our answer to this

The cube root of -512 is -8

And we are done

And you can verify this

Multiply -8 times itself 3 times

Well lets just do it

-8 times -8 times -8

-8 times -8 is positive 64

You multiply that times negative 8

You get -512