Question

Find all the answers of the following: (-256)^(1/4)

Answer 1

i can get to \[4\sqrt[8]{-1} \] but don't know where to go from there to get the solutions

Answer 2

so you know that 4*4*4*4=256

and you know again too that i*i=-1

can you continue it ?

Answer 3

I think he already did that.

Isn't it just e^(ipi/4)?

Answer 4

how did you convert it to the e format

Answer 5

firstly there is (-256)^(1/4 =((-1)^(1/4))*256^(1/4)

true ?

Answer 6

It's deMoivre/Euler formula

(but maybe I have to go all the way round the circle, let me check it)

Answer 7

or continued this ((-1)^(1/2))^(1/2 =i^(1/2)
- so this is the first part with ,,i"

Answer 8

If you get to\[(i ^{2}4^{4})^{\frac{ 1 }{ 4 }}\] then you would do the part inside the brackets and get it to mod-arg form and then bring the 1/4 in using De Moivre. But I can't convert the inside into mod-arg as I don't know what to do with the i^2

Answer 9

Ignoring the 256, which is just 4, then

-1 = cos((2k+1)pi) + isin((2k+1)pi) k = any integer
Take the 4th root
(-1)^(1/4) = cos((2k+1)pi/4) + isin((2k+1)pi/4)
Put k=0,1,2,3

Answer 10

so there are (i^2 *4^4 )^(1/4) so than you can separate these too and will get
(i^2)^(1/4) *(4^4)^(1/4) =i^(1/2) *4

ok ?

Answer 11

@estudier : what do you mean the 256 just goes away and it's 4? how did you get to the first line of your answer?

Answer 12

http://en.wikipedia.org/wiki/De_Moivre%27s_formula#Applications

Answer 13

@jiteshmeghwal9 what you have said there in your reply so this sign that a^(1/2) =a^(-2)
what is equal 1/a^2

how do you think this is right sure ?

Answer 14

For the 4, I am using the sqrt convention so sqrt 256 is 16 and then sqrt 16 is 4

Answer 15

thanks for the wiki link, but, how do i get the r and theta value from -256 or (i^2)(4^4)

Answer 16

oops! sorry . i forgot about that ;P

Answer 17

If you plot the values k = 0 to 3 (ie cos pi/4 + i sin pi/4, etc) on an Argand diagram, then it is at pi/4 + k*pi/2....

The radius is either 1 or 4 depending you want to consider 1 or 256....

Answer 18

i still have no idea how to convert it to what you're saying because you aren't telling me HOW, you're just telling me the answer

Answer 19

I just use the formula......

Answer 20

\[\sqrt[4]{-256}=+- 4i\]

Answer 21

normally questions like this are given in cartesian form i.e. (1+sqrt(3)i)^4 do from that form you can figure out what the angle is and what the modulus is and then use De moivre to solve. But in this example it's in some weird form where the i is being squared so i don't know what to do so just using formulas isn't helping i need to know how to get from this weird form to a normal cartesian form

Answer 22

answer is 4i and -4i

Answer 23

Where is the i being squared?

Answer 24

sorry i think it is only 4i , got that -4i by mistake

Answer 25

\[\sqrt{-1}=i\] use this

Answer 26

@akash_809 : \[\sqrt[4]{-256} = \sqrt[4]{256} \sqrt[4]{-1}\] it's "quad"-rooted not square rooted

Answer 27

yes thats what i have done 4^4 =256 isn't it

Answer 28

4rt (-256) = 4rt (256) 4rt(-1) = 4 4rt(-1) and then use the formula...

Answer 29

@remnant , 4 is fourth root of 256

Answer 30

@estudier what is rt

Answer 31

rt is root

Answer 32

yes 4th root of 256 is 4 but 4th root of -1 is not i

Answer 33

I am just typing it as 4th root.

akash, we have the solution already, we are just explaining it now....

Answer 34

@remnant , roots of negative numbers is not defined and it's imaginary represemted by i and thats what iota means i=sqrt(-1) ...

Answer 35

@akash_809 please read the thread before you post any more....

Answer 36

@estudier i failed to spot the answer in the messages above so gave the answer

Answer 37

@akash_809 Yes, the wrong answer.....

Answer 38

lol

Answer 39

ok... @estudier is the answer something other 4i...i think it should only be 4i..if i m wrong pls correct me

Answer 40

btw hope the picture is clear, it makes everything work in one snap step

Answer 41

\[ \large \left( -1\right)^{1 \over 4} = e^{i{{ \pi +2n \pi \over 4 }}} \text{ where n=0,1,2,3}\]

Answer 42

@experimentX right, that's the Euler half of the puzzle

Answer 43

forget the 256, find -1 on the unit circle, take 1/4 of the angle and then divide the unit circle into 4 parts,
multiply your results by 4

Answer 44

cmon thats the general answer you are giving when cosx= -1 , then we say it as pi or 2n+1 pi...choice is yours...but saying x=pi won't be wrong

Answer 45

@satellite73 been there, done that, its up above....:-)

Answer 46

yes, i see, i was just adding the visual.

Answer 47

i think giving the answer in euler's form is just complicating it...we could simply say its 4i, but then again individulas choice, as i said for cos x=-1

Answer 48

this would be a equally spaced multiple roots ... see the occurrence of roots

Answer 49

@satellite73 and I would give you a medal if I still could....:-)

Answer 50

lol ... i would too

Answer 51

since when is \((4i)^4=-256\)?
thanks i have lots it seems
still can't redeem them for valuable merchandise though

Answer 52

@satellite73 bragging :)