Animated Solution for Mathematics - Binomial Theorem: The sum of all rational terms in the expansion of $(2+\sqrt{3}{)}^8$ is

A Journey into $(2+\sqrt{3}{)}^8$

Welcome, my dear students. Today, we are not just solving a problem; we are peeling back the layers of the Binomial Theorem.

When you look at an expression like $(2+\sqrt{3}{)}^8$, it is easy to feel overwhelmed by the sheer size of the expansion. But in the world of JEE Advanced, we don't fear the expansion; we master it.

We look for the hidden patterns, the structural elegance that allows us to bypass the brute force of manual calculation.

The General Term

Every binomial expansion is governed by a single, powerful DNA sequence: the general term formula. For any expansion of the form $(a+b{)}^n$, the general term $T_{r+1}$ is given by:

Think of this formula as your master key. It allows us to peek into any specific position in the expansion without having to write out all nine terms.

In our case, we have $n=8$, $a=2$, and $b=\sqrt{3}$. Substituting these values, we get:

This is the raw structure. It is the skeleton of our expansion. Now, we must apply our filter.

Why $r$ Matters

We are tasked with finding the sum of all rational terms. This means we need to identify which terms in this expansion are free from the shackles of irrationality.

Look closely at our general term. The binomial coefficient $\left(\genfrac{}{}{0ex}{}{8}{r}\right)$ is always an integer, and the term $2^{8-r}$ is also always an integer. The only potential source of irrationality is $(\sqrt{3}{)}^r$.

For the entire term to be rational, $(\sqrt{3}{)}^r$ must be rational. We know that $\sqrt{3}=3^{1/2}$, so $(\sqrt{3}{)}^r=3^{r/2}$.

For this to be a rational number, the exponent $r/2$ must be an integer. This implies that $r$ must be an even number.

Since $r$ ranges from $0$ to $8$ in our expansion, the valid values for $r$ are $0,2,4,6,$ and $8$. These are the only indices where the irrationality vanishes, leaving us with pure, clean rational numbers.

Calculating the Rational Terms

Now that we have identified our targets, let us calculate them with precision. We have five terms to evaluate:

For $r=0$:

For $r=2$:

For $r=4$:

For $r=6$:

For $r=8$:

We have arrived at the final stage of our journey. We have isolated the rational components of this massive expansion.

All that remains is to sum them up:

Adding these carefully, we arrive at our final result: $18817$.

Take a moment to appreciate what you have just done. You didn't just calculate a sum; you analyzed the structure of a mathematical expression and filtered it based on its properties.

This is the essence of JEE Advanced preparation. It is not about memorizing formulas; it is about understanding the 'why' behind the math.

Whenever you face a problem involving roots in binomial expansions, remember this: the exponent is the key. It is the gatekeeper of rationality. Keep this logic in your toolkit, and you will be ready for any variation the examiners throw at you.