B3. Calculate the value of the expression without using a pocket calculator

$$
\left(\frac{1}{\sqrt{3}-\sqrt{2}}+\frac{2}{\sqrt{8}+\sqrt{12}}\right): \frac{1}{\sqrt{3}}
$$
B3. We rationalize the denominator of the first fraction $\mathrm{v}$ in the parentheses $\frac{\sqrt{3}+\sqrt{2}}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}=\sqrt{3}+\sqrt{2}$ and partially root the denominator of the second fraction $\sqrt{8}+\sqrt{12}=2 \sqrt{2}+2 \sqrt{3}$. We factor out the common factor in the denominator of the second fraction and simplify. We rationalize the second fraction $\frac{\sqrt{2}-\sqrt{3}}{(\sqrt{2}+\sqrt{3}) \sqrt{2}-\sqrt{3})}=$ $\sqrt{3}-\sqrt{2}$. Finally, we add $\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}=2 \sqrt{3}$ and divide $2 \sqrt{3}: \frac{1}{\sqrt{3}}=2 \sqrt{3} \cdot \sqrt{3}$. We get the result 6.

Rationalization of the first fraction $\frac{\sqrt{3}+\sqrt{2}}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}=\sqrt{3}+\sqrt{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1$ point Partial root of the denominator of the second fraction: $\sqrt{8}+\sqrt{12}=2 \sqrt{2}+2 \sqrt{3} \ldots \ldots \ldots .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-13.jpg?height=80&width=1641&top_left_y=217&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-13.jpg?height=57&width=1641&top_left_y=291&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-13.jpg?height=71&width=1639&top_left_y=336&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-13.jpg?height=52&width=1636&top_left_y=391&top_left_x=276)