Question

27z^3-8=0

Answer 1

27x^3=8

Answer 2

x^3=8/27

Answer 3

This time take the cube root.

Answer 4

Not sure how to do that.

Answer 5

\[\sqrt[3]{\frac{ 8 }{ 27 }}\] It looks like that. A square root is when you multiply the same number together twice, a cube root is when you multiply the same number together 3 times.

Answer 6

So what number do you multiply 3 times to get 8 and 27?

Answer 7

7

Answer 8

Okay lets start with 8. Find the prime factorization of 8

Answer 9

2,4,1,8

Answer 10

No, you're just listing the factors. Do you know how to find the prime factorization of a number? @user2486 do you know another way to word this?

Answer 11

Not sure at all, how to do this, sorry.

Answer 12

Seen this before?



prime factors are circled, so 8 = 2*2*2 = 2^3

Answer 13

...

Answer 14

It's alright c: Guessing you've never had to do that before?

Answer 15

No.

Answer 16

Alright all he's doing is writing out the factors of 8. If a number is prime, you leave it alone. If a number is not prime, you write out its factors until all you are left with are prime numbers.
In this case two factors of 8 are 2 and 4. 2 is prime so we leave it alone. 4 can be factored to 2*2. Which cannot be factored any further.
Now that we have all our prime numbers we know the 8=2*2*2
Making any sense?

Answer 17

I guess

Answer 18

Okay so what number multiplied by itself three times makes 8?

Answer 19

2

Answer 20

Okay, so we have \[\sqrt[3]{\frac{ 2*2*2 }{ 27 }?}\] can you figure out what number multiplied by itself three times makes 27?

Answer 21

7

Answer 22

If anyone has a better way to word this, go ahead c:

Answer 23

I'm really bad at multiplication

Answer 24

is it 9?

Answer 25

9X3=27 ya

Answer 26

Okay 7*7=49 49*7=343 so it can't be that.
3*9=27
but 9 isn't prime, what are two factors of nine?

Answer 27

:_ I donz know

Answer 28

3,1,9

Answer 29

?

Answer 30

Okay gimme two numbers that multiply to equal 9.

Answer 31

33

Answer 32

Okay so we know 3*3*3=27
Do you see that?

Answer 33

no

Answer 34

I mean yes

Answer 35

okay so what do we have now

Answer 36

z=2/3

Answer 37

Perfect! You understand why?

Answer 38

Because... I think I might have done it before maybe.

Answer 39

Okay but do you understand why that is the answer?

Answer 40

sure.

Answer 41

Maybe.

Answer 42

Not really.

Answer 43

\(\bf \Large {
27z^3-8=0 \implies z^3 = \cfrac{8}{27} \implies z = \sqrt[3]{\cfrac{8}{27}}\\
27 = 3\times3\times3\\
8 = 2\times2\times2\\
z = \sqrt[3]{\cfrac{8}{27}} \implies z = \sqrt[3]{\cfrac{2\times2\times2}{3\times3\times3}} \implies z = \sqrt[3]{\cfrac{2^3}{3^3}} } \)

Answer 44

\[a^3-b^3=\\
(a - b)(a^2 + a*b + b^2)\\
27z^3-2^3 = ((3z)^3-2^3) = (3z-2)((3z^2)+3*2+4)=0\]

Answer 45

Ok ok I gets it

Answer 46

difference in cubes.....

Answer 47

@trumpetmaster7777, @zzr0ck3r's method will find the imaginary number solution as well, since a cube root has three total roots (one real, two imaginary).