Question

can someone help me understand how to do this I have 3 then a radical -216

Answer 1

\[\sqrt[3]{-216}\]

Answer 2

That 3 is the index. It means to take the 3rd root of -216. It means, "What number makes this sentence true":
\[(?)^3=-216\]

Answer 3

The value of that expression is some number x, such that x*x*x = -216.

Answer 4

Because taking the third root is the inverse of raising to the third power.

Answer 5

oooo, so can I use prime factorization for this?

Answer 6

Here's a hint: \[\sqrt[3]{-8} = -2\]

Answer 7

That might be helpful.

Answer 8

Am I suppose to write and guess until I get something that hits 216? Because this seems ineffective

Answer 9

Yes, sort of. First of all, we know it has to be an even number because an odd number raised to a power could not be even. Secondly, we know it has to be negative.

Answer 10

ok

Answer 11

So, pick up your calculator and start with -2 and raise it to the third power. Do you get -216?

Answer 12

no I dont want to get in the habit of using a calculator and the teacher doesnt like it either

Answer 13

Ok. Now try -4. What is -4 times -4 times -4?

Answer 14

You could also factor 216:

216=2*108 = 2*2*54 = 2*2*2*27 = 2*2*2*3*9 = 2*2*2*3*3*3

Answer 15

how does that help me find out what number cubed is 216

Answer 16

Well, for every group of 3 identical factors, you keep one. Multiply what's left together.

For example, 27 = 3*9 = 3*3*3. We've got 3 3's, so we keep one. Well, 3*3*3 = 27, so 3^3 = 27.

Answer 17

ok I get that part

Answer 18

How about finding the cube root of 64? 64 = 2*2*2*2*2*2. We can write that as (2*2*2)*(2*2*2) and if we keep 1 out of every 3, that means 2*2 = 4 is the cube root of 64.

Answer 19

How about 1000? 1000 = 2*500 = 2*2*250 = 2*2*2*125 = 2*2*2*5*25 = 2*2*2*5*5*5. You've got 3 each of 2 and 5. Keep 1 of each, multiply them together, 2*5 = 10. 10^3 = 1000. 10 is the cube root of 1000. Get the picture?

Answer 20

Handy rules for determining divisibility when factoring:

if the number is even, it is divisible by 2
if the number ends in 5 or 0, it is divisible by 5
if the sum of the digits is a multiple of 3, it is divisible by 3

Answer 21

hmm ok I think I get what your saying I got a little lost when u got to the higher numbers but thats cause my mind just shut down let me look back at what you said

Answer 22

oooo ok I think I got it.

Answer 23

It's just like for square root, except instead of needing pairs of identical factors, you need triples...

Answer 24

so let me try another one

Answer 25

How about 512?

Answer 26

u want the cube index? or squared

Answer 27

cube root of 512, since that's what you were doing in this problem.

Answer 28

ooo I got stuck because I was factoring and I thought it had to factor evenly but I got it so 8 cubed is 512

Answer 29

Correct!

Answer 30

hmmm ok so I was trying for 5 radical -243 and I got stuck look at what I did