Powers and roots

Can you produce convincing arguments that a selection of statements about numbers are true?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

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.

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

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?

Find the values of $\sqrt{2 + \sqrt 3}- \sqrt{2 - \sqrt3}$ and of $\root 3 \of {2 + \sqrt 5}+ \root 3 \of {2 - \sqrt 5}$

Find all the solutions to the this equation.