# Surds

**What is a surd?**

When the **square root of a number is a decimal number** (and not a whole number), it is called a **surd**.

Examples of surds are:

- 2 = 1.4
- 3 = 1.7
- 5 = 2.2
- 6 = 2.4
- 7 = 2.6

**What is NOT a surd?**

When a **square root of a number is a whole number**, it is **not a surd**.

Examples that are not surds are:

- 1 = 1
- 4 = 2
- 9 = 3

## Maths Fun - Compass Activity

- Draw a right-angled triangle. Make each of the sides that form the right angle 1 unit long.
- Using Pythagoras's Rule, the hypotenuse is 2.
- Extend a compass to the length of 2 and draw the next right-angled triangle with a side of 1 and the second side 2.
- Using Pythagoras's Rule, the hypotenuse of the next triangle is 3.
- Continue this to find the square roots of whole numbers.
- Consider that the ancient Greeks probably used this method to work square roots of surds and non-surds. Clever, isn't it?

## Questions - Which are Surds?

Find the square roots of these numbers - 1, 2, 3, 4, 5, 25, 50, 100.

Which are surds?

**Answer**

Surds are the square roots of 2, 3, 5, 50.

## Example One - Simplify Surds

Simplify:

(a) 8

(b) 27

(c) 48

(d) 50

**Answers:**

Firstly, find the factors of the number where **one of the factors is a square number** such as 4, 9, 16, 25 and so on. Then simplify.

(a) 8 = 4 × 2 = 22

(b) 27 = 9 × 3 = 33

(c) 48 = 16 × 3 = 43

(d) 50 = 25 × 2 = 52

## Questions - Simplify these Surds

(a) 12

(b) 45

(c) 500

(d) 75

**Answers**

(a) 23

(b) 35

(c) 105

(d) 53

## Example Two - Entire Surds

Make these entire surds:

(a) 32

(b) 25

(c) 35

(d) 57

**Answers:**

This is the reverse of Example One above.

(a) 32 = 9 × 2 = 18

(b) 25 = 4 × 5 = 20

(c) 35 = 9 × 5 = 45

(d) 57 = 25 × 7 = 175

## Questions - Make Entire Surds

(a) 53

(b) 210

(c) 37

(d) 73

**Answers**

(a) 75

(b) 40

(c) 63

(d) 147

## Example Three - Simplify these Multiplied Surds

Simplify these surds:

(a) 23 × 53

(b) 25 × 52

**Answers:**

(a) 23 × 53

= 4 × 3 × 25 × 3

= 900

= 30

(b) 25 × 52

= 4 × 5 × 25 × 2

= 1000

= 100 × 10

= 1010

## Example Four - Simplify these Divided Surds

Simplify these surds:

(a) | 85 |

25 |

(b) | 67 × 53 |

23 |

**Answers:**

(a) | 85 | = | 8 | = 4 |

25 | 2 |

(b) | 67 × 53 | = | 67 × 5 | = | 37 × 5 | = | 157 |

23 | 2 |

## Questions - Simplify these Surds

(a) 23 × 53

(b) | 43 |

26 |

**Answers**

(a) 30

(b) 2

## Example Five - Simplifying Surds on the Bottom of a Fraction

Simplify these surds:

(a) | 3 |

2 |

(b) | 5 |

3 |

**Answers:**

If there is a surd on the bottom of a fraction, multiply both the top and the bottom of the fraction by that surd. Then simplify.

(a) | 3 | = | 32 | = | 32 |

2 | 2 × 2 | 2 |

(b) | 5 | = | 53 | = | 53 |

3 | 3 × 3 | 3 |

## Questions - Simplify these Surds

(a) | 7 |

2 |

(b) | 8 |

2 |

**Answers**

(a) | 7 2 |

2 |

(b) 42