New roots for old...
Or, how to produce cube and other roots from square ones ...

A basic calculator usually has a key for square roots, but not for higher ones, so, if you want, say, the cube root of 5, how can you get it from your calculator?

Roots can be written as powers; the square root of 3, for example, can be written as 3^(1/2), and the cube root of 5 as 5^(1/3). How can we use this information with our basic calculator?

This is where a knowledge of number-bases comes in handy. In base two one-third can be expressed as 0·01 01 01 01 ..., so we can write the cube root of a as a^(0·01010101....)

This in turn can be expanded as
a^(0·01+0·0001+0·000001+...), which is
a^0·01 times a^0·0001 times a^0·000001 times... etc

Now, in base two, a^0·1is the square root of a, and a^0·01 is the fourth root of a, or the square root of (the square root of a). We write √a for "the square root of a", and we can write a^0·01 = √(√x) etc.

So I can calculate the cube root of, say, 5, by calculating:

a=√(√5); and store the value of a
b=√(√a); and store the value of b
c=√(√b); and store the value of c
and so on, finally approximating to the cube root of 5 as:

a, a*b, a*b*c, a*b*c*d and so on. (Using * to mean "multiply")

For other roots you will need to write down the binary representation of the fraction:
for example, one-fifth = 0·0011 0011 0011 ..., one-seventh = 0·001 001 001 ...

(Of course, if you have a scientific calculator handy you can work out higher roots automatically and more efficiently; when I developed the method described above there were few scientific calculators around, and they were very very expensive.)

I must be having a déjà vu, because I was convinced this thread already existed (I mean, previously to October 22nd, 2007). :huh: Or at least I seem to remember having already read this same method of approximating any root in a calculator from square roots by using the binary representation of the fractional exponent somewhere.

It's quite possible that I've posted it twice - rather forgetful nowadays (they call it having a "senior moment")! I had it ready in html form for the DSGB site but hadn't posted it there.
The only other place it was published was in a mathematical magazine - many years ago - but I can't remember the name of the magazine.