How do calculators calculate fractional and decimal exponents, and how could it be done on paper?

Question

I've recently been intrigued by calculating decimal and fractional powers. An example is $5^{3.5}$. I can break this into $5*5*5*5^{1/2}$, but how would I express this or find the answer without using square roots? I'm not very experienced in mathematics and haven't taken precalc yet, so I asked my teacher. She said it's impossible without square roots, but if that were true calculators wouldn't be able to get the answer. What steps would I have to take to try to solve this without using square roots?

Answer 1

The most likely method that calculators would use for exponentiation is via a logarithmic transformation. For example, to calculate $5^{3.5}$, the calculator would do something resembling the following:

1. Find $\ln 5 \approx 1.609$

2. Calculate $3.5 \times 1.609 = 5.6315$

3. Calculate $e^{5.6315} \approx 279.08$

To calculate the logarithms and exponentials, there are many methods available (see, e.g. this question), but one way is to use Taylor series approximations:

$\begin{eqnarray} \ln (1 + x) & = & x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \\ e^x & = & 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \ldots \end{eqnarray}$

and there are tricks to making these converge very quickly so the calculator only needs to perform a few operations to get a high accuracy.

This is not too different to how people used to perform these calculations before electronic or even mechanical calculators were common. They would either use a slide rule, which used logarithmic scales to automatically perform steps 1 and 3 so that the addition of lengths results in a multiplication of values, or would reference a table of logarithmic values that were pre-computed.