Calculating $a^p+b^p$ given $a^2+b^2=c$ If one has $a^2+b^2=c$ where $a$,$b$ and $c$ are real numbers, is there any way to calculate $a^p+b^p$ where $p$ may be any real number? If there is, would you please explain with an example with $p=3$?

The other responses showing you cannot find it just from $a^2+b^2=c$ are correct.  You may be interested in the fact that given $a^2+b^2=c, a+b=d$, you can form $(a^2+b^2)(a+b)=cd=a^3+a^2b+ab^2+b^3$ and  $d^2-c=2ab$, so $cd-\frac{d^3-cd}{2}=a^3+b^3$.  Given as many symmetric polynomials as variables you can find all the higher orders.
