Question

\[\LARGE (-1)^{\frac{5}{9}}\]
Doesn't equal to 1 or -1??

http://www.wolframalpha.com/input/?i=%28-1%29%5E%285%2F9%29

Answer 1

\[\LARGE
(-1)^{\frac{5}{9}}=(-1)^{\frac{50}{90}}=\sqrt[90]{(-1)^{50}}=\sqrt[90]{1}=1
\]

Answer 2

Wolfram|Alpha says no: http://www.wolframalpha.com/input/?i=%28-1%29%5E%285%2F9%29

Googles says no: http://puu.sh/MFqd

Calculator says -1.

Answer 3

?????

Answer 4

something to do with taking even root of a negative number

Answer 5

\[\LARGE \sqrt[9]{-1^5}=\sqrt[9]{-1}\]

Answer 6

-1

Answer 7

because num and den are odd

Answer 8

\[-1^{5/9}=\sqrt[9]{-1^5}=\sqrt[9]{-1}=-1^{1/9}=-1\]
that means that when you multiply -1 nine times by itself you will get -1
think of the cubic root of -27

Answer 9

(-1)^(odd/odd) = -1

Answer 10

yes parity will help here

Answer 11

my calculator says math error.. :|

Answer 12

(-1)^(even/odd) = 1
(-1)^(even/even) = 1
(-1)^(odd/even) = i

Answer 13

Wolfram said otherwise!

Answer 14

wolf ram says its -0.17

Answer 15

\[\huge \sqrt[90]{1^{50}} = \pm 1\]

Answer 16

i mean \[\huge \sqrt[90] 1 = \pm 1\]

Answer 17

im confused

Answer 18

@Igbasallote i dont think that's correct..!!

Answer 19

\[\huge (-1)^{90} = 1\]
\[\huge(1)^{90} = 1\]

Answer 20

\[\large \pm1\]..
1 can't satisty \(\LARGE \sqrt[9]{-1}\)

Answer 21

exactly my response

Answer 22

\[\huge (-1)^{\frac 59} \ne (-1)^{\frac{50}{90}}\]

Answer 23

nope

Answer 24

thats even/even

Answer 25

oh wait...

Answer 26

?????????????? (-1)^(5/9) = -1

Answer 27

Are you telling me \(\LARGE \frac{5}{9}\ne\frac{50}{90}\)???????

Answer 28

sort of

Answer 29

odd/odd is not even/even

Answer 30

im telling you that raising it to 5/9 is different from raising it to 50/90

Answer 31

So, \(\LARGE 5^{\frac{1}{1}}\ne5^{\frac{10}{10}}\)

Answer 32

yes it is

Answer 33

show me what you do to (-1)^(5/9) to make it (-1)^(50/90)

Answer 34

you cant play around with it like its a ratio because its not, its something raised to a ratio. not the same thing

Answer 35

just because 1 out of 2 cars are mine does not mean 10 out of 20 are:)

Answer 36

\[\LARGE (-1)^{\frac{5}{9}}=(-1)^{\frac{5}{9}*1}=(-1)^{\frac{5}{9}*\frac{10}{10}}=(-1)^{\frac{50}{90}}\]And I can play around with the exponent, it's just a fraction!

Answer 37

ok i guess its broke then?

Answer 38

you will run into the same problem with any odd/odd you change to even/even

Answer 39

I seem to be getting a variety of results for some of these,..
1. (-1)^(5/9) is a complex number according to both Wolfram and Google
2.. (-1)^(1/9) is -1 according to Google, but not Wolfram (Complex)
3. (-1)^5 is -1 for both Google and Woflram
4. ((-1)^5)^(1/9) is -1, according to Google, but not Woflram (Complex)
5. ((-1)^(1/9))^5 is -1 according to Google, but not Wolfram (Complex)
6. ((-1)^(1/90)) is a complex number again by both Google and Wolfram
7. ((-1)^(50))^(1/90) is 1 according to both Woflram and Google.

(3.), (6.), and (7.) were expected... but the others are interesting.
I'd love to hear some explanation for these, if it's like some complex number magic that I never learned or something. D:

Answer 40

\[(-1)^{\frac{5}{9}} \\= e^{\frac{5}{9}\;\ln(-1) } \\= e^{i\frac{5\pi }{9}} \text{ applying Euler's formula } \\= \cos\left(\frac{5\pi}{9} \right)+i\sin\left(\frac{5\pi}{9} \right)\]

Answer 41

more generally \[(-1)^n = \cos\left(n\pi \right)+i\sin\left(n \pi \right),\quad n\in \mathbb{C}\] I think