Powers and roots

Simple additions can lead to intriguing results...

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.

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

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

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

Find all the solutions to the this equation.

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

Find the sum of this series of surds.