how to raise a number to the power of 0.x?

The simplest way to do it is like this.

Say you want to work out 3^0.4

Turn the power into a fraction: 0.4 = 2/5

So 3^0.4 = 3^(2/5)

The numerator of the fraction is the power and the denominator is the root, so you want to find the 5th root of (3^2).

Finding 3^2 is easy. It's 3x3 = 9.

Now you need the 5th root of 9. The only easy way to do this without a calculator is by trial and error, like this:

You want to find a number which can be multiplied by itself 5 times to get 9. Make a reasonable guess, say 2:

2x2x2x2x2 = 32 (too high)

Try 1:

1x1x1x1x1 = 1 (too low)

So you know the answer is between 1 and 2. Try 1.5:

1.5 x 1.5 x 1.5 x 1.5 x 1.5 = 7.59 (too low)

So you know the answer is between 1.5 and 2. Try 1.7

1.7 x 1.7 x 1.7 x 1.7 x 1.7 = 14.20 (too high)

So you know the answer is between 1.5 and 1.7. Try 1.6

And so on.

The actual answer to 3^0.4 given by a calculator is 1.55, so you can see you're getting closer and closer. You keep going with this until you have the answer to the number of decimal places you want. It's long winded but it works. There are other methods as some people have said, but these are all advanced mathematics.

2^.2 = 2^(1/5) or the fifth root of 2. There is no simple way to find this without the use of logarithms, infinite series, or Newton's Method of roots. You could use a log(base e) table in a book. Example: 2^.2=e^(.2*ln 2)

ln 2=.6932 and (.2)*ln 2=.1386

e^x=1 + x + x^2/2 + x^3/3! + x^4/4! ... Where n!=1*2*3...*n.

Insert 0.1386 in for x and get 1.148. You could also look it up in the table.

5^0.17= e^(.17*ln 5) = e^(.17*1.609) = e^(.2735) = 1.315

4^0.14= e^(.14*ln 4) = e^(.14*1.3863) = e^(.1941) = 1.214

Just remember that a^b = e^(b*ln a)

Let's say you put a number to the power of 0.5 you can square it to get the original number this is because x^(b/a) = ª√(x^b) - it is probably easier to see written. Anyway since a number to the power of 0.x is the same as a number raised to the power of a fraction you will get a fraction of the number. It is hard to explain but you can't really use simple operations, you could use the [x√] button but it has to be the reciprocal ([1/x] or [x^(-1)] button) of the decimal.