Surds

Key Properties of Surds

Multiplication Rule for Square Roots:

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}, \text{ where } a, b \geq 0$

This property allows us to split a square root into separate roots of its factors.

Division Rule for Square Roots:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \text{ where } a, b > 0$

This property allows us to separate the square root of a fraction into a ratio of square roots.

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Worked Examples:

Simplifying Using the Multiplication Rule:

$$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$

Simplifying Using the Division Rule:

$$\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4}$$

Simplify a Combination of Surds:

Simplify $\sqrt{72}$

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$

Important Note:

These properties are not in the log tables. As an exercise, try proving them using the rules of indices:

For $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, use:

$$a^{1/2} \times b^{1/2} = (ab)^{1/2}$$

For $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, use:

$$(a/b)^{1/2} = a^{1/2} \times b^{-1/2}$$

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Summary:

• A surd is a root expression that cannot be simplified into a rational number.
• Key properties:
  1. $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
  2. $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
• Use these rules to simplify surds into their simplest forms.