## Introduction

A surd is an irrational number that can be represented as the root of a number. It’s essentially a number that can’t be simplified to remove the root. The most common example of a surd is the square root of a number that is not a perfect square.

## Definition

A surd is an expression containing a root where it cannot be simplified further. For example, \(\sqrt{2}\) and \(\sqrt{3}\) are surds because they can’t be simplified any further.

## Properties of Surds

### 1. Multiplication:

For all non-negative numbers a and b:

\(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)

### 2. Division:

For all non-negative numbers a and b, where b ≠ 0:

\(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

### 3. Rationalizing the Denominator:

To rationalize the denominator containing a surd, you often multiply the numerator and denominator by a conjugate. For instance, consider:

\(\frac{1}{\sqrt{2}}\)

To rationalize, multiply by the conjugate, \(\sqrt{2}\):

\(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \sqrt{2}/2\)

### 4. Addition and Subtraction:

Like terms can be added or subtracted:

\(\sqrt{a} + \sqrt{a} = 2\sqrt{a}\)

\(3\sqrt{b} – 2\sqrt{b} = \sqrt{b}\)

However, unlike terms cannot be combined:

\(\sqrt{a} + \sqrt{b}\) remains as it is.

## Simplifying Surds

Surds can be simplified by breaking down the number under the root into its prime factors.

### Example:

To simplify \(\sqrt{18}\):

Factor 18 into its prime factors: \(18 = 2 \times 3 \times 3\) or \(2 \times 3^2\).

Therefore,

\(\sqrt{18} = \sqrt{2 \times 3^2} = 3\sqrt{2}\)

Of course, I’ll present examples followed by their solutions for each rule:

### Examples for Surd Rules:

**Multiplication of Surds**:

a) Multiply \(\sqrt{2}\) and \(\sqrt{8}\).

Solution: \(\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4\)

b) Multiply \(\sqrt{3}\) and \(\sqrt{3}\).

Solution: \(\sqrt{3} \times \sqrt{3} = \sqrt{9} = 3\)

c) Multiply \(\sqrt{5}\) and \(\sqrt{10}\).

Solution: \(\sqrt{5} \times \sqrt{10} = \sqrt{50} = 5\sqrt{2}\)

d) Multiply \(\sqrt{4}\) and \(\sqrt{6}\).

Solution: \(\sqrt{4} \times \sqrt{6} = \sqrt{24} = 2\sqrt{6}\)

**Division of Surds**:

a) Divide \(\sqrt{18}\) by \(\sqrt{2}\).

Solution: \(\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{9} = 3\)

b) Divide \(\sqrt{50}\) by \(\sqrt{5}\).

Solution: \(\frac{\sqrt{50}}{\sqrt{5}} = \sqrt{10}\)

c) Divide \(\sqrt{27}\) by \(\sqrt{3}\).

Solution: \(\frac{\sqrt{27}}{\sqrt{3}} = \sqrt{9} = 3\)

d) Divide \(\sqrt{8}\) by \(\sqrt{4}\).

Solution: \(\frac{\sqrt{8}}{\sqrt{4}} = \sqrt{2}\)

### Examples for Rationalization:

**Single Term in Denominator**:

a) Rationalize \(\frac{4}{\sqrt{3}}\).

Solution: \(\frac{4\sqrt{3}}{3}\)

b) Rationalize \(\frac{5}{\sqrt{7}}\).

Solution: \(\frac{5\sqrt{7}}{7}\)

**Binomial with a Surd in Denominator**:

a) Rationalize \(\frac{2}{1 + \sqrt{3}}\).

Solution: \(\frac{2(1 – \sqrt{3})}{-2} = \sqrt{3} – 1\)

b) Rationalize \(\frac{3}{2 – \sqrt{5}}\).

Solution: \(\frac{3(2 + \sqrt{5})}{-1} = -6 – 3\sqrt{5}\)

c) Rationalize \(\frac{4}{\sqrt{2} + \sqrt{3}}\).

Solution: Multiply by the conjugate \(\sqrt{2} – \sqrt{3}\) to get \(\frac{4(\sqrt{2} – \sqrt{3})}{-1} = 4\sqrt{3} – 4\sqrt{2}\)

d) Rationalize \(\frac{5}{\sqrt{7} – 1}\).

Solution: Multiply by the conjugate \(\sqrt{7} + 1\) to get \(5\sqrt{7} + 5\)

## Concluding Remarks

Surds form an essential concept in algebra, especially in quadratic equations and geometry. Understanding their properties can greatly simplify algebraic expressions and solve problems that involve radicals.

Remember to always look for ways to simplify a surd, and remember that surds have particular rules that differ from standard algebraic operations!