$$\sqrt{8}=\sqrt{4·2}=\sqrt{4}·\sqrt{2}=2\sqrt{2}$$

$$\frac{\sqrt{18}}{\sqrt{2}}=\frac{\sqrt{9·2}}{\sqrt{2}}=\frac{\sqrt{9}·\sqrt{2}}{\sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{2}}=3$$

$$\sqrt{\frac{75}{3}}=\frac{\sqrt{5^2·3}}{\sqrt{3}}=\frac{5\sqrt{3}}{\sqrt{3}}=5$$

$$\sqrt{18a^2}=\sqrt{9·2·a^2}=3a\sqrt{2}$$

$$\sqrt{8}·\sqrt{2}=\sqrt{16}=4$$

$$=\sqrt{2·8}=\sqrt{16}=4$$

$$=\sqrt{3·12}=\sqrt{36}=6$$

$$\sqrt{\frac{32}{2}}=\sqrt{16}=4$$

$$\frac{\sqrt{54}}{\sqrt{6}}=\sqrt{9}=3$$

$$\sqrt{18}-\sqrt{2}=\sqrt{9·2}-\sqrt{2}=3\sqrt{2}-\sqrt{2}=2\sqrt{2}$$

$$=\frac{9·\sqrt{3}}{\sqrt{3}·\sqrt{3}}=\frac{\cancel{3}·3\sqrt{3}}{\cancel{3}}=\frac{3\sqrt{3}}{1}=3\sqrt{3}$$

$$=\frac{6·\sqrt{2}}{\sqrt{2}·\sqrt{2}}=\frac{\cancel{2}·3·\sqrt{2}}{\cancel{2}}=3\sqrt{2}$$

$$=\frac{a·\sqrt{a}}{\sqrt{a}·\sqrt{a}}=\frac{\cancel{a}\sqrt{a}}{\cancel{a}}=\sqrt{a}$$

$$=\frac{x·\sqrt{x}}{2\sqrt{x}·\sqrt{x}}=\frac{\cancel{x}·\sqrt{x}}{2·\cancel{x}}=\frac{\sqrt{x}}{2}=\frac{1}{2}\sqrt{x}$$

$$=\sqrt{9·2}=\sqrt{9}·\sqrt{2}=3\sqrt{2}$$

$$=\sqrt{4·3}+\sqrt{3}=\sqrt{4}·\sqrt{3}+\sqrt{3}=2\sqrt{3}+\sqrt{3}=3\sqrt{3}$$

$$=\sqrt{49·2}-\sqrt{36·2}=\sqrt{49}·\sqrt{2}-\sqrt{36}·\sqrt{2}=7·\sqrt{2}-6\sqrt{2}=\sqrt{2}$$

$$=\sqrt{25·3}-5\sqrt{3}=\sqrt{25}·\sqrt{3}-5\sqrt{3}=5\sqrt{3}-5\sqrt{3}=0$$