Negative Fractional Powers

Date: 12/08/98 at 23:37:02
From: amanda nelson
Subject: Negative square roots

My question is about negative square roots. I was wondering what 3 with 
-2/3 as the power would be. I know that 27 with -2/3 is 3 to the third, 
so the threes cancel and it becomes 3 with -2 as the power that equals 
1/9, but otherwise I am stumped. Any help would be greatly appreciated.

Thank you,
Amanda

Date: 12/09/98 at 03:19:00
From: Doctor Pat
Subject: Re: Negative square roots

Amanda,

First, break the problem into two parts, or three when you have 
fractions. To start, a negative power is just a special instruction 
that says to take the reciprocal of the number before doing the powers 
or roots: 
 
   3^-2 = (1/3)^2  

Then, if you have a fractional power, x^(a/b)  it means raise x to the 
a power, then take the bth root of the answer. You can always reverse 
the order of these two if it is easier and take the root and then the 
power. So problems like 27^(-2/3) really have three parts:
 
   1) the negative which says take the reciprocal of the base:

      27^(-2/3) = (1/27)^(2/3)
         
   2) the root is easier so we take the third root of (1/27), which 
      is 1/3

   3) the square of 1/3 will be 1/9   

Here is an example in which the base number itself is a fraction:

   (9/16)^(-3/2)   Take the negative away first.
   (16/9)^(3/2)    Now do the root (square root).
   (4/3)^3         Then the power.
    64/27          And we are done. 
   
If you do the problems in three steps like this, they should start to 
seem "do-able" in a few trys.  

Good luck,

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