# How to raise a number to a power geometrically.

There are methods to add two lines of arbitrary lengths or multiply them together known since Greek times; and more advanced methods based on the concepts of bases and units.

But, I have not been able to find a way to exponentiate a number geometrically without using algebra. I would love if someone could somehow illustrate the concept.

Basically I am asking is it possible to draw the graph of a^x geometrically.

On questions raised by Aretino and RickyDemer I want to clarify that: I am talking about Euclidean geometry (so a collapsible compass,straight-edge are allowed); although, Cartesian geometry is fine, too.

Also, is there a book that can teach a basic concept as this? You know, a book on Euclidean geometry that teaches exponentiation, multiplication etc.

• For integer exponents, you could consider $a^n$ as the volume of an $n$-dimensional cube with side lengths of $a$ – Christian Aug 28 '16 at 16:29
• @Christian What about fractional exponents? And, how exactly can I illustrate the concept on n-dimensional cube in Euclidean geometry? – Abdur Rahman Aug 28 '16 at 16:32
• do you mean to construct a number $a^n$ from a given segment of length $a$ and $1$? – gambler101 Aug 28 '16 at 16:33
• – user57159 Aug 28 '16 at 16:35
• He's talking about the theorem on the page my initial comment linked to. ​ ​ – user57159 Aug 28 '16 at 16:57

Let $AB = 1$, $AD=a$. We draw another line at any angle with $AB$, mark a point C on it such that $AC=a$. Let a line through $D$ parallel to $BC$ meet this line at $E$, then $AE=a^2$, continuing this way we can raise it to any integer power.
Using this you can only calculate negative integral powers as well. Calculation of square roots is simple so we can construct all numbers of form $a^\frac{n}{2}$. But we cannot go cube roots or anything as using scale and compass we are limited to square roots.