Animated Solution for Mathematics - Complex Numbers: If $z^2+z+1=0$, where $z$ is complex number, then the value of $(z+1/z{)}^2+(z^2+1/z^2{)}^2+(z^3+1/z^3{)}^2+\cdots +(z^6+1/z^6{)}^2$ is

When you first encounter the equation $z^2+z+1=0$, do not just see it as a quadratic equation. See it as a gateway. This is the defining equation of the cube roots of unity.

If you multiply both sides by $(z-1)$, you get $z^3-1=0$, which tells us that the roots are the complex numbers $\omega$ and ${\omega }^2$. Imagine standing on the Argand plane; these roots are not random points. They are the vertices of an equilateral triangle inscribed in the unit circle.

We need to evaluate the expression:

We know that for these roots, ${\omega }^3=1$. This is our golden key. It implies that:

Let us apply this to the first term where $k=1$:

We also know that $1+\omega +{\omega }^2=0$, which means $\omega +{\omega }^2=-1$. So, the first term is simply $(-1{)}^2=1$.

For the second term where $k=2$:

For the third term where $k=3$:

We have found the values for the first three terms: $1,1,4$. Because ${\omega }^3=1$, the powers of $\omega$ are periodic with a period of $3$.

This means that for $k=4,5,6$, the values will be identical to $k=1,2,3$. The fourth term is $1$, the fifth term is $1$, and the sixth term is $4$.

The total sum is simply:

The final result is 12. By understanding the soul of the equation—the periodicity of the cube roots of unity—we turned a massive summation into a simple, rhythmic pattern.