Calculating Irrational and Fractional Powers of a Number by Hand

Question

We can write $3^{0.6}$ as $3^\frac6{10}$ and can be simplified to $3^\frac3{5}$, yet it is tedious to calculate by hand, as well as $3^e$ and $3^{\pi}$? How can I do this without calculator? Like how can such things be calculated by hand?

Answer 1

In your example since $0.6$ is a rational number, you can express it as $\frac{3}{10}$. But $e$ and $\pi$ are examples of irrational numbers so you cannot express them as a fraction. The best approach is to use the series expansion of $a^b = e^{b\ln a}$ as explained in the comments to compute an estimate of the numerical value to a desired degree of accuracy.

But if you just want to express it in a form where the exponent is some fraction as in your example then, although you cannot exactly represent $e$ or $\pi$ as a fraction due to their irrationality, you can always find rational approximations of irrational to any desired degree of accuracy using the continued fractions. Then you can express it is

$$ 3^{\pi} = 3^{\frac{a_n}{b_n}} + \text{an error term depending on $n$} $$

where $\frac{a_n}{b_n}$ is a convergent, $n = 1,2,3,\ldots$. The bigger is $n$ the better is your the estimate $3^{\frac{a_n}{b_n}}$ and the smaller is the error is error term. For example the first few convergents of the continued fraction of $\pi$ are

$$ 3, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102},\ldots $$

Most well known constants have well known continued fractions but in general finding the continued fraction of an arbitrary irrational number involves numerically estimating the number to several decimal places so at end of the day, this method is not any more efficient then directly estimating the value using the series expansion of $e^t$.

Answer 2

If you are skilled in written computation, you can form a table of successive square roots, starting from $3$ (i.e. exponents $\frac12,\frac14,\frac18\cdots$). This can be done using the direct method for square roots (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Decimal_(base_10)), or by Newton's iterations.

Now write the power as a fractional binary number and compute the product of the corresponding powers.

$$\begin{align}1&\to 3\\\tfrac12&\to1.7321\\\tfrac14&\to1.3161\\\tfrac18&\to1.1472\\\tfrac1{16}&\to1.0711\\\tfrac1{32}&\to1.0349\\&\cdots\end{align}$$

Note that the next roots are obtained by just halving the fractional part.

$$\pi=11.0010010000\to 3^\pi\approx3^3\cdot1.1472\cdot1.0173=31.5103$$

(Only the first decimal is correct :-( )