Animated Solution for Mathematics - Complex Numbers: If $a$ and $b$ are real numbers such that $(2+\alpha {)}^4=a+b\alpha$, where $\alpha =\frac{-1+i\sqrt{3}}{2}$, then $a+b$ is equal to

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving an algebraic expression; we are peeling back the layers of a beautiful mathematical structure.

When you first look at the expression $(2+\alpha {)}^4$ with $\alpha =\frac{-1+i\sqrt{3}}{2}$, it is natural to feel a bit intimidated. It looks like a brute-force expansion waiting to happen.

But in the world of JEE Advanced, brute force is rarely the intended path. There is almost always a secret key, a hidden symmetry waiting to be discovered.

Look closely at $\alpha =\frac{-1+i\sqrt{3}}{2}$. Does it ring a bell? This is the complex cube root of unity, commonly denoted as $\omega$.

This number is special because it satisfies two powerful properties:

These aren't just equations; they are tools that allow us to simplify complex powers into simple, manageable linear forms. By identifying $\alpha$ as $\omega$, we have already won half the battle.

We need to evaluate $(2+\omega {)}^4$. Instead of jumping into a fourth-power expansion, let's be strategic. We can view this as the square of a square: $((2+\omega {)}^2{)}^2$.

Let's focus on the inner term first. Expanding $(2+\omega {)}^2$ gives us:

Now, here is where the magic happens. We don't want that ${\omega }^2$ term hanging around. Using our identity $1+\omega +{\omega }^2=0$, we can rewrite ${\omega }^2$ as $-1-\omega$.

Substituting this back in, we get $4+4\omega +(-1-\omega )$, which simplifies beautifully to $3+3\omega$, or $3(1+\omega )$.

Now, we take our simplified inner term and square it:

We are almost there! We just need to expand $(1+\omega {)}^2$, which is $1+2\omega +{\omega }^2$. Again, we invoke our trusty identity ${\omega }^2=-1-\omega$.

Substituting this into our expression, we get $1+2\omega +(-1-\omega )$, which simplifies down to just $\omega$. So, our entire expression $(2+\omega {)}^4$ collapses into $9\omega$.

We were given the form $a+b\alpha$. Since we found that $(2+\alpha {)}^4=0+9\alpha$, we can perform a direct comparison.

It is clear that $a=0$ and $b=9$. The final step is simply to find the sum $a+b$, which gives us:

Isn't it satisfying? What started as a daunting fourth-power expansion turned into a graceful dance of identities.

Remember, in complex numbers, whenever you see powers of $\omega$, look for ways to reduce them using ${\omega }^3=1$ and $1+\omega +{\omega }^2=0$. Keep practicing this mindset, and you will find that even the most complex problems start to reveal their simple, elegant cores. You have the tools—now go forth and conquer!