Roots and Coefficients

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?

Three by One

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

Target Six

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

Cube Roots

Age 16 to 18 Challenge Level:

Evaluate without a calculator:

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