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?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
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?
Looking first for real roots, note that
There will be altogether 9 complex values of
of Madras College found the
complex values and dicovered some beautiful patterns when he
plotted them in the complex plane. Neil's discoveries can be
generalised to a 1/3
- b 1/3
for any real or complex numbers a and b,
and from cube roots to n th
In order to find patterns similar to the
ones discovered by Neil, but in a simpler situation, and to see how
his ideas can be generalised, you may like to plot the twelve
values of 8 1/3 + 81 1/4
in the complex plane.