Surds

A surd is an irrational number that can be expressed as a root that cannot be simplified into a rational number. Typically, surds involve square roots or higher roots that are not perfect squares (or cubes, etc.). \(\sqrt{2}​ ​\text{ , and } \sqrt{3}\) are surds because their exact decimal values are irrational.

Simplifying Surds

Surds can often be simplified by factoring out perfect squares.

\[\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\]

Adding and Subtracting Surds

Surds can only be added or subtracted if they have the same radicand (the number inside the square root).

\[3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\]

but:

\[\sqrt{2} + \sqrt{3} \quad \text{cannot be simplified further.}\]

Multiplying and Dividing Surds

To multiply or divide surds, use the following rules:

Multiplication: Multiply the numbers inside the square roots.

\[\sqrt{2} \times \sqrt{3} = \sqrt{6}\]

Division: Divide the numbers inside the square roots.

\[\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2\]