Surd Equations and Properties Summary

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

1. \[ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \]

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

2. \[ \sqrt{a} + \sqrt{b} \quad \text{cannot be simplified unless} \quad a = b \]

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

3. \[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]

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

4. \[ \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \]

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

5. \[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \]

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

6. \[ (\sqrt{a})^2 = a \]

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

7. \[ \frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b} \]

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

   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}