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### 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...

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/
```
Associated Topics:
High School Calculators, Computers

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