The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is:

Step 1: Identify the general term

We use the binomial theorem to find the general term of the expansion $(2+\sqrt{3})^8$. Here, $n=8$, $a=2$, and $b=\sqrt{3}$. The general term is given by the formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.

Step 2: Substitute values into the general term

Substituting the values $n=8$, $a=2$, and $b=\sqrt{3}$ into the general term formula, we get $T_{r+1} = \binom{8}{r} (2)^{8-r} (\sqrt{3})^r$. We can simplify $(\sqrt{3})^r$ as $3^{r/2}$.

Step 3: Determine conditions for rational terms

For the term $T_{r+1}$ to be rational, the power of $\sqrt{3}$ must be an even integer. This means $r$ must be an even integer, since $r$ can range from $0$ to $8$. So, possible values for $r$ are $0, 2, 4, 6, 8$.

Step 4: Calculate rational terms for each 'r'

Now we calculate each rational term by substituting the even values of $r$ into the general term formula. We find $T_1$ (for $r=0$), $T_3$ (for $r=2$), $T_5$ (for $r=4$), $T_7$ (for $r=6$), and $T_9$ (for $r=8$).

Step 5: Sum the rational terms

Finally, we sum all the calculated rational terms to find the total sum of all rational terms in the expansion.

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