07-16-2014, 07:48 PM (This post was last modified: 07-16-2014 07:53 PM by jebem.)
Post: #11 jebem Senior Member Posts: 1,343 Joined: Feb 2014
(07-16-2014 06:01 AM)John R Wrote:
(07-16-2014 02:08 AM)Waon Shinyoe Wrote:  (-8)^(2/3)=4

Careful -- fractional exponentiation of negative numbers is a thorny issue, and the answer is different depending on the assumed domain and the precise definition of the exponentiation operator (which was not specified here). Although 4 is typically the accepted answer in the real domain, a more general interpretation (based on analytic continuation of exponentiation in the complex domain) is that there are THREE answers of the form $(-8)^{2/3}= \{4\omega, 4\omega^2, 4\},$ where $\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ is the so-called principal cube root of unity. (Try evaluating $$\omega^3$$ if you don't believe it.) The first of these three answers, $4\omega = -2 + i2\sqrt{3},$ would generally be considered the principal answer.

If you don't believe me, consult a higher authority: try evaluating (-8)^(2/3) on an HP-42S and see what it says. Interesting... and my HP-15C gives me an Domain Error.
I mean, using -8^( 2/3)

Jose Mesquita