1. Zadatak

U razvoju binoma

$$\left(\sqrt[3]{\frac{2x}{\sqrt y}}-\sqrt{\frac{y}{2\sqrt[3]{x}}}\right)^{21}$$

opći je član $\binom{21}{k}a^{21-k}b^k$, gdje je

$$a=\sqrt[3]{\frac{2x}{\sqrt y}}=2^{1/3}x^{1/3}y^{-1/6}, \qquad b=-\sqrt{\frac{y}{2\sqrt[3]{x}}}=-2^{-1/2}x^{-1/6}y^{1/2}.$$

Sada je

$$\begin{aligned} \binom{21}{k}a^{21-k}b^k &=\binom{21}{k} \left(2^{1/3}x^{1/3}y^{-1/6}\right)^{21-k} \left(-2^{-1/2}x^{-1/6}y^{1/2}\right)^k\\ &=\binom{21}{k}(-1)^k 2^{\frac{21-k}{3}-\frac{k}{2}} x^{\frac{21-k}{3}-\frac{k}{6}} y^{\frac{k-21}{6}+\frac{k}{2}}. \end{aligned}$$

Kako bi $x$ i $y$ imali isti eksponent vrijedi

$$\frac{21-k}{3}-\frac{k}{6}=\frac{k-21}{6}+\frac{k}{2}.$$

Odavde dobivamo $k=9$, pa je traženi član

$$-\binom{21}{9}\cdot2^{-1/2}x^{5/2}y^{5/2}.$$