### 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?

### Target Six

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

### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Cube Roots

### Why do this problem?

It provides experience of using algebra and working with surds. It leads into complex cube roots if you want to explore the patterns arising with those solutions.

### Key questions

We have to evaluate

$c^{1\over 3} - d^{1\over 3}$ what are $c^3$, $d^3$, $cd$ and $(c-d)^3$?