Decimal Exponents

Date: 03/19/99 at 11:06:39
From: Richard Urich
Subject: Decimal Exponents

Is there a way to find x^y, where y is a decimal, without the use of 
a calculator? I know x^(y/z) can be expressed as (x^y)^(1/z), but I 
cannot figure out how to do a (1/z) exponent without a calculator. It 
has to be possible if the calculator does it, so I would like to know 
how calculators do it.

Thank you.

Date: 03/19/99 at 18:25:38
From: Doctor Rick
Subject: Re: Decimal Exponents

When I was in school, we did not have calculators, but we could 
compute x^y. We had another help, though: a logarithm table (or a 
slide rule, which is a log table on a stick). Somebody spent a 
lifetime calculating the logarithms, and I could use that person's 
lifetime work to calculate powers.

You can do the same on a calculator without touching the x^y key. For
instance, to find 3.14^2.718, you would enter:

  3.14 LOG * 2.718 = 10^x

getting the answer 22.42099, which is the same answer I get using the 
x^y key. In math notation,

  log(x^y) = y * log(x)

I do not know, but I assume that calculators use their log function to
compute powers. I also do not know for sure how calculators implement 
the log and exponential functions. There are a number of series 
expansions that could be used to approximate the log as closely as 
needed, but I would not be surprised if they use some method 
incorporating a lookup table.

If you want to find out about how calculators do it, you might try an
article referenced in this Dr. Math Archives answer:

  http://mathforum.org/dr.math/problems/kuhn12.3.96.html   

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