Powers and roots

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

What have Fibonacci numbers got to do with Pythagorean triples?

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Find the smallest value for which a particular sequence is greater than a googol.

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.