I haven't studied complex numbers in great depth myself. To me it can still seem mind-blowing that the exponent operator can output many values at the same time ?. But, I think I may have just had an epiphany while writing this. The exponent operator can output
infinite values if we write \(3^x\) because x can be any value. And you wouldn't think twice if you saw \(3^x=9\). Therefore it shouldn't be so surprising that \(1^{(1/2)} \) could output two values and we sometimes need to consider both. The equals = "operator" only needs one match from LHS to RHS before it's a true statement (more than one match can, of course, be possible). I don't know if this is a good way of thinking? Perhaps one of the more experienced helpers will know!
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(Probably because I've been involved with programming for years) I often find it helpful to think of ONLY the principal root, or
principal value, being returned from the exponent operator. Programs like "Matlab" just return the principal root if you type (-1)^(0.2) So, if you're sticking to principal roots, then it's worth noting that the "power of power" rule:-
[math] \left(a^b\right)^c = a^{(bc)} [/math]
with a,b,c real, can only be applied when one of the following is true:-
- a>0
- a<0 and c integer (b can be ANY real)
- a=0 and b>0 and c>0
You wrote...
[math] \left(-1\right)^{2/10} = \left(\left(-1\right)^2\right)^{\color{red}1/10} [/math]
this doesn't work when using principal roots because c, highlighted red, isn't integer. However, this would be fine...
[math] \left(-1\right)^{2/10} = \left(\left(-1\right)^{1/10}\right)^2 [/math]