2/3 Power using a Calculator

Date: 07/05/2001 at 11:49:34
From: Tan Jia Hui
Subject: Indices

Hi,

I'm currently facing this particular problem of whether -512 can be 
the answer to the equation x^(2/3) = 64. When I use my calculator, it 
simply gives error. Can you explain? 

Thanks a lot!

Regards, 
Tan Jia Hui

Date: 07/05/2001 at 12:41:10
From: Doctor Peterson
Subject: Re: Indices

Hi!

The problem is that the calculator is not as smart as you and I; it 
doesn't know any math. It has to convert 2/3 to a decimal before 
using it, and x^0.66666666 does not exist for a negative number, 
though x^(2/3) does. Such a power only exists for rational exponents 
whose denominator is odd, and 0.666666666 doesn't simplify to an odd 
denominator.

Since you know that you want the 2/3 power exactly, you can just 
square -512 and then take the cube root; or take the cube root of -512 
and then square it. In fact, try using your calculator this way:

    ((-512)^2)^(1/3)
or
    ((-512)^(1/3))^2

Mine can do both of these. I'm not sure how it manages to do the 
latter; perhaps it recognizes 1/3 as a special case.

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