Here is a list of common equations and properties related to surds:

1. Simplifying surds:

   $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$

   Example: (\sqrt{2} \times \sqrt{3} = \sqrt{6})

2. Adding and subtracting surds:

   $$\sqrt{a} + \sqrt{b} \quad \text{cannot be simplified unless} \quad a = b$$

   Example: (\sqrt{2} + \sqrt{2} = 2\sqrt{2})

3. Multiplying and dividing surds:

   $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$

   Example: (\frac{\sqrt{6}}{\sqrt{2}} = \sqrt{3})

4. Rationalizing the denominator (multiplying by the conjugate):

   $$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$$

   Example: (\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2})

5. Square of a binomial involving surds:

   $$(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$$

   Example: ((\sqrt{2} + \sqrt{3})^2 = 2 + 3 + 2\sqrt{6} = 5 + 2\sqrt{6})

6. Squaring a single surd:

   $$(\sqrt{a})^2 = a$$

   Example: ((\sqrt{5})^2 = 5)

7. Rationalizing a binomial (denominator with surds):

   $$\frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$$

   Example: (\frac{1}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7})

8. Cube roots and simplifying:

   $$\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}$$

   Example: (\sqrt[3]{4} \times \sqrt[3]{2} = \sqrt[3]{8} = 2)

Latex

\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}
\sqrt{a} + \sqrt{b} \quad \text{cannot be simplified unless} \quad a = b
\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}
(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}
(\sqrt{a})^2 = a
\frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}
\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}