Understanding Exponent Rules: The Confusion of (-8)^(2/6)

[flame_m13]

hello.i have a weird problem.

(-8) ^ (2/6) = ?

my first instinct was to simplify the exponent, which yields (-8) ^ (1/3) = -2.
this is the answer the calculator gives as well.
but someone pointed out if you don't simplify that exponent, you have the sixth root of (-8^2), which would give you positive 2. they mentioned something about the symbolic proof of squares...

i always thought you would simplify first, but maybe I'm wrong?

this is confusing.

 

[NoMoreExams]

How about

-1 = i^2 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1? :)

 

[snipez90]

Hmm I can see the confusion that you see. I think simplifying it is correct in this case though I have tried looking at it in other ways. Well first of all, if you try to take the 6th root, it will obviously be undefined. Similarly, you can't just square because you'll get a "loss of information" which results in 2, which I think is incorrect provided that simplifying first is the correct way.

Now I did try splitting it into (-1)^(1/2)(2/3)(8)^(2/6). Unfortunately, this leads to more or less worse problems. If I try anything but take the square root of -1 first in the first term of the product, I would get 2 as an answer. Of course taking the square root of -1 first would take this problem into the realm of imaginary numbers. Perhaps someone else could give a better reason for simplifying first.

 

[snipez90]

How about

-1 = i^2 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1? :)

That kind of manipulation is forbidden. You can't just multiply like that. $$\sqrt{-1}\sqrt{-1} \neq \sqrt{(-1)^2} = \sqrt{1}$$

In fact I think to use the property $$\sqrt{a}\sqrt{b} = \sqrt{ab}$$, at least one of a, b must be positive.