Non-integer Powers and Exponents

Date: 01/06/99 at 10:53:41
From: Christina 
Subject: Taking a number to a power that's not an even number

I understand x squared or x cubed, but how do you get x to the 1.9 for 
instance?

Date: 01/06/99 at 13:11:15
From: Doctor Rob
Subject: Re: Taking a number to a power that's not an even number

Thanks for writing to Ask Dr. Math!

The expression x^(1/n) is defined to be the nth root of x. If you are
comfortable with finding the nth root of a number, the rest follows.
Then x^(m/n) = [x^(1/n)]^m, so you can compute it as the mth power of
an nth root, where m and n are positive whole numbers.

Take the exponent you are interested in, such as 1.9, and write it as
a fraction

   1.9 = 19/10

Then

   x^(19/10) = [x^(1/10)]^19

In the case of a terminating decimal, you can always use a denominator
which is a power of 10. In the case of a repeating decimal, you can
always use a denominator that is a power of 10 times one less than
another power of 10:  10^a*(10^b-1), for some integers a and b.

In the case of an irrational number like sqrt(2) = 1.4142..., you can
approximate x^sqrt(2) by x^1, then x^1.4, then x^1.41, then x^1.414,
then x^1.4142, and so on, until two successive approximations agree to
as many decimal places as you need.

Example: Find 5^sqrt(2) to five decimal places of accuracy.

   5^1 = 5.0000000000
   5^1.4 = 5^(14/10) = [5^(1/10)]^14 = 1.7461894309^14 = 9.5182696936
   5^1.41 = 5^1.4*5^(1/100) = 9.5182696936*1.0162245913 = 9.6726997289
   5^1.414 = 5^1.41*5^(4/1000) = 9.67269973*1.006458519 = 9.7351710392
   5^1.4142 = 9.7383051743
   5^1.41421 = 9.7384619075
   5^1.414213 = 9.7385089280
   5^1.4142135 = 9.7385167647
   5^1.41421356 = 9.7385177051
   5^1.414213562 = 9.7385177365

So to five decimal places, 5^sqrt(2) = 9.73852.

There is another way using logarithms and antilogarithms which can cut
down the amount of numerical computation, but requires knowledge of 
math through calculus to explain in detail.

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