Powers and roots

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Explore the relationships between different paper sizes.

Find the sum of this series of surds.

What does this number mean? Which order of 1, 2, 3 and 4 makes the highest value? Which makes the lowest?

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?

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

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.