1. Zadatak

Racionalizirajmo najprije nazivnike zadanih razlomaka.

$$\frac3{\sqrt[3]4-\sqrt[3]2+1} =\frac3{\sqrt[3]4-\sqrt[3]2+1}\cdot\frac{\sqrt[3]2+1}{\sqrt[3]2+1} =\frac{3(\sqrt[3]2+1)}{2+1}=\sqrt[3]2+1$$ $$\frac{\sqrt2}{2+\sqrt2} =\frac{\sqrt2}{2+\sqrt2}\cdot\frac{2-\sqrt2}{2-\sqrt2} =\frac{2\sqrt2-2}{4-2}=\frac{2(\sqrt2-1)}2=\sqrt2-1.$$

Uvrstimo dobivene vrijednosti u početni brojevni izraz i izračunajmo.

$$\begin{aligned} \sqrt2+\left(\frac3{\sqrt[3]4-\sqrt[3]2+1}-1\right)^3-\frac{\sqrt2}{2+\sqrt2} &=\sqrt2+(\sqrt[3]2+1-1)^3-(\sqrt2-1)\\ &=\sqrt2+(\sqrt[3]2)^3-\sqrt2+1\\ &=\sqrt2+2-\sqrt2+1\\ &=3. \end{aligned}$$