TITLE: What is $(-8)^\frac{2}{3}$?
QUESTION [6 upvotes]: I am comfuse about something.
I want to compute $(-8)^\frac{2}{3}$
Is it $(-2^3)^\frac{2}{3}$=$(-2)^{3\cdot\frac{2}{3}}$=$(-2)^2=4$ ?
Is there any problem here because the base is negative?
Thanks.

REPLY [3 votes]: Please have a look at:
How do you compute negative numbers to fractional powers?
With rational exponents for negative numbers you are in trouble.
You can do $x^{1/3}$ but if you try the thing you want then there are problems. You find that it
depends on the order how you perform the operations resp. which root you choose.
Therefore if we consider only real numbers, such things as $(-8)^{2/3}$ are not well defined. See also:
http://en.wikipedia.org/wiki/Exponentiation#Rational_exponents
here is the example from this site:
$−27 = (−27)^{((2/3)⋅(3/2))} = ((−27)^{2/3})^{3/2} = 9^{3/2} = 27$