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:

$$\sqrt{2}+\sqrt{3}\text{cannot be simplified further.}$$

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.

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