3. Find the sum

$$
\begin{aligned}
& \frac{1}{(\sqrt[4]{1}+\sqrt[4]{2})(\sqrt{1}+\sqrt{2})}+\frac{1}{(\sqrt[4]{2}+\sqrt[4]{3})(\sqrt{2}+\sqrt{3})}+ \\
& +\ldots+\frac{1}{(\sqrt[4]{9999}+\sqrt[4]{10000})(\sqrt{9999}+\sqrt{10000})}
\end{aligned}
$$
Answer: 9. Solution. If we multiply the numerator and denominator of the fraction $\frac{1}{(\sqrt{n}+\sqrt{n}+1)(\sqrt{n}+\sqrt{n+1})}$ by $\sqrt[4]{n}-\sqrt[4]{n+1}$, we get $\sqrt[4]{n+1}-\sqrt[4]{n}$. The specified

![](https://cdn.mathpix.com/cropped/2024_05_06_94cb46a707e666e84118g-2.jpg?height=68&width=1586&top_left_y=215&top_left_x=324)
$\sqrt[4]{10000}-\sqrt[4]{1}=9$