Resolving Decimal Exponents

Date: 03/26/2001 at 19:04:22
From: Adel
Subject: Powers

Please explain how to find 7^.3 or 5^.6 without using a calculator, 
and give me the formula.

Thanks,
Adel

Date: 03/27/2001 at 12:58:58
From: Doctor Rick
Subject: Re: Powers

Hi, Adel.

It's not easy, but we can at least think about how we might do it, and 
be glad we don't actually have to do it.

Let's take 7^0.3 as our example. Write the decimal as a fraction:

     7^(3/10)

We can write 3/10 as 3 times 1/10. The multiplication property of 
exponents tells us that this can be written as:

     7^(3 * 1/10) = (7^3)^(1/10)
or
     7^(1/10 * 3) = (7^(1/10))^3

Look at the second form: 7^(1/10) is the tenth root of 7. You can find 
methods for finding the square root and even the cube root of a number 
in the Dr. Math Archives, but they are methods of successive 
approximation. That is, you make a "guess" and use the method to find 
a better "guess," then use it again to find an even better "guess," 
etc. The number of steps it takes depends on how accurate you want 
your answer to be.

Here is one approximation method (called Newton's Method) for finding 
the nth root of a number y:

     x_2 = x_1 * (1-1/n) + y/n/x_1^(n-1)

You can start with any guess for x_1 - say, 1 - and this formula will 
give you a number x_2 that is a better approximation to the nth root 
of y. Then put this new number in place of x_1 in the same formula, 
and you'll get a third number that is an even better approximation. 
With n = 10 and y = 7, you'll get the answer 1.214814 (accurate to six 
decimal places) in seven steps. It's tedious without a calculator (I 
used a spreadsheet, which amounts to the same thing), but it can be 
done.

Once you have the tenth root of 7, then you have to raise this result 
to the third power to get the answer to your problem.

     1.214814 * 1.214814 * 1.214814 = 1.792789

Another approach is to use the properties of logarithms. In 
particular,

     log(7^0.3) = log(7) * 0.3

You can use printed log tables or a slide rule instead of a calculator 
to find the logarithm of 7, which is 0.84509804. Multiply this by 0.3 
to get 0.253529412. Then you must use the log table in reverse, 
finding the number whose logarithm is 0.253529412. It turns out to be 
1.79278996, which (as we just saw) is 7^(3/10). 

This method is easier than the first, but it does require something 
other than paper and pencil (and begs the question, "How are 
logarithms calculated?").

Anyway, those are two ways of calculating a number to a decimal power.

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