Al calcular las operaciones $\left ( \frac{18\times 25}{2^{3}\times 3^{1}\times 5^{2}} \right )^{\frac{1}{2}}$ el resultado es

 $$\left ( \frac{18\times 25}{2^{3}\times 3^{1}\times 5^{2}} \right )^{\frac{1}{2}}$$

$$=\left ( \frac{2\times 3^{2}\times 5^{2}}{2^{3}\times 3^{1}\times 5^{2}} \right )^{\frac{1}{2}}$$

$$=(2^{1-3}\times 3^{2-1}\times 5^{2-2})^{\frac{1}{2}}$$

$$=(2^{-2}\times 3^{1}\times 5^{0})^{\frac{1}{2}}$$

$$=2^{-2\times {\frac{1}{2}}}\times 3^{1\times {\frac{1}{2}}}\times 5^{0\times {\frac{1}{2}}}$$

$$=2^{-1}\times 3^{\frac{1}{2}}\times 5^{0}$$

$$=\frac{3^{\frac{1}{2}}}{2}$$

$$=\frac{\sqrt{3}}{2}$$

Al efectuar las operaciones $\left ( \frac{3^{-2}\times 0.3\times 10}{2^{-3}\times 0.2\times 20} \right )^{2}$ el resultado es

$$\left ( \frac{3^{-2}\times 0.3\times 10}{2^{-3}\times 0.2\times 20} \right )^{2}$$

$$=\left ( \frac{\frac{1}{3^{2}}\times \frac{3}{10}\times 2\times 5}{\frac{1}{2^{3}}\times \frac{2}{10}\times 2^{2}\times 5} \right )^{2}$$

$$=\left ( \frac{\frac{2\times 3\times 5}{2\times 3^{2}\times 5}}{\frac{2^{3}\times 5}{2^{4}\times 5}} \right )^{2}$$

$$=\left ( \frac{\frac{1}{3}}{\frac{1}{2}} \right )^{2}$$

$$=\left ( \frac{2}{3} \right )^{2}$$

$$=\frac{4}{9}$$