Decimal / Rational Exponents BY HAND

[EebamXela]

I'm reviewing for a test. One of my questions on the review (and incidentally a question I've had in my own mind for a long time) is how do you evaluate an ARBITRARY rational exponent with pencil and paper and no calculator?

The specific problem i was givens is "Solve without a calculator: 3.4$$^{2.1}$$ "

I know the expression can be broken up to look like this:

3.4$$^{2}$$ * 3.4$$^{0.1}$$

But how in the world can you calculate the tenth root of something on paper.

Or perhaps I'm missing some critical trick or rule or something. Please help.

Can you only practically use Newton's method?

 

[dodo]

Probably this won't help you, but what people in the old days (before calculators) would do, is to convert the expression to
$$3.4^{2.1} = \exp(2.1 \times \log(3.4))$$
and then look up logarithms and antilogarithms in a table like http://en.wikipedia.org/wiki/Handbook_of_Mathematical_Functions_with_Formulas,_Graphs,_and_Mathematical_Tables" .

 

[Unit]

This thread may be of some help.