Question

If z^2+1/z^2=11 , find the value of z^3-1/z^3,using only the positive value of z-1/z

Answer 1

Hello,

Let's assume

$t=z- \dfrac{1}{z} \\\\ t^2=z^2-2+ \dfrac{1}{z^2}= 11-2=9==\ \textgreater \ t=3\\\\ t^3=z^3-3z^2* \dfrac{1}{z} +3z* \dfrac{1}{z^2} - \dfrac{1}{z^3} \\\\ =z^3 - \dfrac{1}{z^3} -3(z-\dfrac{1}{z})\\\\ z^3 - \dfrac{1}{z^3}=t^3+3t=3^3+9=27+9=36$