Fractional Exponents

Date: 03/11/99 at 18:06:19
From: Tom Judd, D.V.M.
Subject: Fractional Exponents

I am a veterinarian at a large goat farm and I am trying to figure 
out the water consumption of the average goat.

One of the formulae that I have is 145 grams of water per kilogram of 
bodyweight to the power of .75 (145g per  BW Kg^.75).

I cannot remember how fractional exponents work!

Our average goat weighs 77 Kg.

I think 77 kg ^.75 would be 

(.75 X 77 kg) X (.75 X 77 Kg) 
  = (57.75 X 57.75) 
  = 3335.06

Have I figured this out correctly?

Thank you!

Date: 03/15/99 at 12:36:31
From: Doctor Mike
Subject: Re: Fractional Exponents

A fractional power relates to the expression of the power in numerator 
and denominator form. You have 3/4 for the 0.75 decimal. Similarly, 
0.5 is 1/2 and 0.3333333 is 1/3 and so on. If you represent the 
fraction as n/d then x to the n/d power means the  d-th  root of 
x to the power n. 
   
Examples: If d is 2 then we deal with the 2nd root, 
             which is commonly referred to as the square root.
          If d is 3 then we deal with the 3rd root, 
             which is commonly referred to as the cube root.
          If d is 4 then we deal with the 4th root, 
             which you get by taking the square root twice.
  
So, for your example of 77 to the 3/4 power, you would first take 77 
to the third power, take the square root of that, and then, take the 
square root again. What we get is 
   
            SQRT( SQRT( 77*77*77 ) )  
          = SQRT( SQRT(  456533  ) )
          = SQRT(     675.67226    )  
          = 25.993697 
  
which is about 26. I think, though, that this is a good place to use 
an 8-dollar pocket calculator. Just enter 77, then the y^x key (or 
something similar), then 0.75, and finally the = key.    
   
Your final answer is 145*25.993697 = 3769.086 grams = 3.769086 kg, or 
a bit under 4 kilograms of water. Of course, I cannot speak for the 
accuracy of the formula.  
   
I hope this helps.

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