What is the value of (-1)^(2/4)?

Question

a) (-1)^(1/2) = +/- i

OR

b) [(-1)^2]^(1/4) = 1^(1/4) = +/- i, +/- 1

Answer 1

There are actually FOUR solutions (notice the FOURth root). In the most basic sense, such a number is defined to be any solution to the equation:

z^4 = (-1)² = 1

That's how we define fractional powers. That means z is a fourth root of unity. There are four:

1
-1
-i
i

Cancellation of exponents is ONLY allowed when you're taking principle exponents - meaning only when you're disregarding the multiplicity of solutions to such equations. Fractional exponents only behave like normal fractions (ie follow cancellation rules, etc) when we make those kinds of single-answer-only assumptions.

For example:

x ^ (2/2) = 1

Only has one solution (obviously) if you cancel the exponent (x = 1). But it has another solution too (x = -1), since:

(-1)^(2/2) = √((-1)²) = √(1) = 1

Finding ALL the solutions (ie not disregarding multiplicities of solutions) is always the convention in questions regarding complex numbers. Things like canceling exponents, taking positive square roots, and "answer does not exist" are conventions of real valued problems and aren't appropriate in addressing this question.

Answer 2

Order of operations says parentheses come before exponents. So the answer is +/- i.

(-1)^(2/4)
Simplify terms in parentheses:
= (-1)^(1/2)
Exponent:
= +/- i

edit: What the heck are you talking about two people below me?

Do you realize that 2/4 = 1/2???

They are the exact same thing. It's like saying 0.999999....repeating infinitely is equal to 1. There are many different ways to say 1/2, but in the end it's simplified form is 1/2. Maybe you should take a course in proof and learn about the rational numbers and how they are defined.

Man, I shouldn't even have to type this crap.

edit 2: cheeser - great explanation. Instead of putting me down, you actually gave a legitimate explanation as to why I am wrong. Thank you.

Answer 3

The answer is very controversial so there is no definite answer (we clearly see that it is controversial....)

The side I would be taking is b).

We can say that in value, 2/4 = 1/2, but 2/4 isn't the actual fraction 1/2.

So using exponent laws, you should have gotten choice b).

What changes everything is that we have an even numerator.

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Or, you can try to make it seem like a) is also equal to +/- 1 by doing the following.

When simplified to the max, a) = b)

a) * b) = 1

Sqrt both sides, but also remember b) = a)

So a) = +/- 1

So mathematically you can show that a) = +/- i and +/- 1

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This also reminds me of something like this.

x = 3

x(x) = 3(x)

x^2 = 3x

x = 0 or 3

I still have to say, there is no definite answer.

It could be that +/-1 might be an extra answer or maybe +/- i is an extra answer.

Depends on how you solved the problem.

Answer 4

2/4 = 1/2, thus

(-1)^(2/4) = (-1)^(1/2) = +-i

(-1)^(2/4) cannot be +-1 because
(+-1)^(4/2) = 1

Answer 5

(-1)^(2/4) = (-1)^(1/2) = sqrt(-1) = ±i

By definition, sqrt(-1) = ±i

Cogito, what are you talking about? The 2/4 was in parentheses, and it can be simplified to 1/2.

Answer 6

Dr. D~~ unless your talking about Euro's your oringinal eqations,of (-1)^(2/4)
(-1)^(1/2)
(-1)^(.500)
-1^.5=-.5
why did y change it? I had to come back and still I am coming up with -1or 3
if your speaking about Euro's then I come up with big lol!!

but still I am coming up with USD 8.00-0.50=7.50
now we are talking dollars,, however I can talk political math if you would like than in that case we would have -1 presidential vote =2/3rd majority in congress,,thats a big lol too ,, however if you want me to talk about the planet skies or in biblical terms,, I will do this too,,,I am having fun with a laught out loud to myself,,,,anywhoo,, have any more,,,or shall I keep going to Newtons law,,or Lewis Carroll, and more,, or just plain different languages.
what about Christian Math, War, Veto's , computer language, or this is why I can not use the computer any more ,, so tired good nite,, and good luck to all~~

Answer 7

Its sad how the first person fails to grasp the depth of your problem.

Whats sadder, though, is how the second person sees a parenthesis in the exponent when there isnt one. You wrote the parenthesis for clarity in typed form... but in reality your problem can just as easily be written without the parenthesis in the exponent and suddenly, magically, person number two would be stumped all over again

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On a side note, isnt there a third possiblity?
[ (-1)^(1/4) ]^2

Answer 8

There is no unique value.
For k=...-2,-1,0,1,2,...
(-1)^(2/4)
=exp(pi*i+2*pi*k)^(2/4)
=exp(pi/2*i+pi*k)
=exp(pi*k*i)*exp(pi/2*i)
=(-1)^k*exp(pi/2*i)
=(-1)^k*i
=+i or -i.