How can I calculate a non-integer power of a number? Integer powers are easy to calculate by repeated application of multiplication. However if a power is not an integer then I always need to use my calculator. How can I calculate a non-integer power without a calculator? 
For example, how does one compute 10 raised to 0.90 without a calculator?
The closest I've come to solving the problem is doing fractional power approximations for upper and lower bounds.  For example 10 raised to the .31 is greater than 10 raised to the 1/2 but less than 10 raised to the one third
10 raised to the 1/2 is 1/100.  10 raised to the 1/3 is 1/1000.  But there is no where near to a good answer................. I think
 A: There is no easy way. Before the invention of calculators people used books of logarithms or slide rulers. To tackle this problem you would note:
$$\log\left(10^{0.9}\right)=0.9\log10$$
You would then look up $\log10$ in your log book and multiple that by 10 before reverse looking up that value to get the answer.
Glossary:
A log book is a large table of values with the logarithm of each value next to it. It would usually be several/many pages as a high degree of accuracy was desired. It would usually include all the logarithm rules and instructions as a preface.
Google Images of Logarithm Books
EDIT: You added some of your own work which has some misconceptions in it:
"10 raised to the 1/2 is 1/100. 10 raised to the 1/3 is 1/1000. But there is no where near to a good answer................. I think"

This is incorrect. 10 raised to the power of -2 is $\frac{1}{100}$ where as 10 raised to the power $\frac{1}{2}$ means $\sqrt{10}$. Similarly 10 raised to the power of -3 is $\frac{1}{1000}$ where as 10 raised to the power $\frac{1}{3}$ means $\sqrt[3]{10}$. This is a common mistake students make in learning their index laws.
A: $10^{0.9}=10^{1-0.1}=\frac{10}{10^{0.1}}$ thus by binomials for small powers $10^{0.1}= 10\times 0.1+2(0.1)=1.2$ thus we can easily calculate $\frac{10}{1.2}=8.33..$ and calculator gives it as $7.94\approx 8$. Hope its clear.
A: Graphically:
Take a sheet of paper and draw an horizontal axis from $0$ to $1$ and vertical axis from $0$ to $10$, $10$ cm long each.
Mark the points $(0,1),(0.5,3.16)$ and $(1,10)$ (as $3.16\approx\sqrt{10}$). You can also mark $(0.3,2)$ and $(0.7,5)$ (as $10^{0.3}\approx2$).
Draw a smooth curve through these points, and you have a nice base $10$ logarithm/exponential calculator. Also switch your mind to scientific notation.

Taking your example, lookup $10$ on the vertical axis and read the logarithm, $1$. Multiply by the exponent $0.9$, giving $0.9$. Lookup $0.9$ on the horizontal axis and read the antilogarithm, $7.9$.
Other example: cubic root of $90$, i.e. $(9\cdot10^1)^{1/3}$ ? The log of $9$ is $0.95$, hence that of $90$ is $1.95$ and dividing by three, $0.65$, which corresponds to $4.5$.
You can expect about $1.5$ exact digit by this method.
A: it is useful, in the context of your specific example, to recall that
$$
5^{10} = 9765625 \\
4^{10} = 1048576
$$
this gives
$$
(1+\frac14)^{10} = \frac{9765625}{1048576}
$$
thus we look for $a$ satisfying:
$$
(1+a)^{10} = \frac{10485760}{9765625} =1 + \frac{720135}{9765625}
$$
retaining powers of $a$ below the third, we require:
$$
10a+45a^2 +\dots=\frac{720135}{9765625}
$$
giving
$$
a = \frac{-10 +\sqrt{100 + 4\times 45\times\frac{720135}{9765625}}}{90} \\
=\frac19\left(\sqrt{1+\frac{1296243}{9765625}}-1\right) 
$$
just to check we are on the right track, note that taking the first two non-cancelling terms in the binomial expansion of the square root, gives $a=0.0071294$ and $(1+a)^{10}(\frac54)^{10} =9.9989\dots$
A: There is a very easy way.
Since 0.9 = $\frac{9}{10}$
Therefore $10^{0.9} = 10^\frac{9}{10} = \sqrt[10]{10^9} = 7.94$ 
And it works for any decimal number.
