Why Don't Calculators Accept (-2)^(2/3)?

Date: 02/17/2002 at 06:06:18
From: Aaron Kim
Subject: Index number

Dear Dr. Math,

   (-2)^(2/3)
      = cubic root of (-2)^2
      = cubic root of 4
      = 1.587

Theoretically the above calculation is right; however, if you punch in 
(-2_^(2/3) in a scientific calculator, it gives an error message. That 
is undefined. Could you explain why?

Best regards.
Aaron.

Date: 02/17/2002 at 08:00:47
From: Doctor Jerry
Subject: Re: Index number

Hi Aaron,

Most calculators evaluate a^b this way:

a^b = e^{ln(a^b)} = e^{b*ln(a)}.

So, if a is negative, your calculator may give an error message - for 
the reason that the domain of the ordinary logarithm function is the 
set of all positive numbers.  

My HP48GX gives

(-2)^{2/3} = -0.7937... + i*1.3747...

for an answer. Your result, 1.587..., is also an answer. The fact is 
that there are several answers. There are three cube roots of 4. If 
you cube -0.7937... + i*1.3747... you will obtain 4 (except for 
rounding error).

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/