If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

There are many different methods to solve this geometrical problem - how many can you find?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Evaluate without a calculator:

$(5\sqrt 2 + 7)^{{1\over 3}} - (5\sqrt 2 - 7)^{{1\over 3}}.$