Question

The sum of the rational terms in the expansion of $(\sqrt{2}+5\sqrt{3}{)}^{10}$ is:

• A. 41
• B. 42
• C. 39
• D. 45

Answer 1

The correct option is A

41

$T_{r+1}$ = 10$C_r\frac{10-r}{2^2}{.3}^{\frac{r}{5}}$

This is rational, if $\frac{10-r}{2}$ and $\frac{r}{5}$ are integers.

$\therefore$ There are only two rational terms.

Namely 10$C_0\left(\sqrt{2}{)}^{10}{\left(3^{\frac{1}{5}}\right)}^{0}and10C_{10}\right(\sqrt{2}{)}^0{\left(3^{\frac{1}{5}}\right)}^{10}$

$\therefore$ sum = 32 + 9 = 41