A surd is a number written in a form that includes a square root. For example \sqrt 3 , 1+\sqrt 2 and 7\sqrt 5 are all examples of surds. A surd is a useful way of giving a value in a precise, or exact, form where there is no need to be concerned with accuracy or decimal places, and therefore a calculator is not needed.

Examination papers often use square numbers in questions on surds, so look out for these numbers \qquad \qquad \qquad 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256...

and simple multiples of them.

Examination papers often use square numbers in questions on surds, so look out for these numbers \qquad \qquad \qquad 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256...

and simple multiples of them.

## Summary/Background

The expression \sqrt{x} means the positive square root of x and is called a surd.

Surds such as \sqrt{2} can be evaluated on a calculator, for example \sqrt{2} = 1.414... \, \,, however this immediately introduces the issue of accuracy. Instead of evaluating, we use some algebraic properties of surds in order to simplify them, for example factors that are square numbers themselves.

Remember the all-important rules:

Surds such as \sqrt{2} can be evaluated on a calculator, for example \sqrt{2} = 1.414... \, \,, however this immediately introduces the issue of accuracy. Instead of evaluating, we use some algebraic properties of surds in order to simplify them, for example factors that are square numbers themselves.

Remember the all-important rules:

- \sqrt{ab} = \sqrt{a}\sqrt{b}
- \displaystyle \sqrt{\frac{a}{b} } =\frac{ \sqrt{a} }{\sqrt{b} }

- \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}
- \sqrt{a - b} \ne \sqrt{a} - \sqrt{b}

## Software/Applets used on this page

## Glossary

### square root

of a number n, that value that when squared equals n

### surd

A number containing one or more irrational square roots.

### union

The union of two sets A and B is the set containing all the elements of A and B.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic |
---|---|---|---|

AP Calculus AB (USA) | 1 | Algebra and Functions | Surds |

AP Calculus BC (USA) | 1 | Algebra and Functions | Surds |

AQA A-Level (UK - Pre-2017) | C1 | Algebra and Functions | Surds |

AQA AS Maths 2017 | Pure Maths | Algebra | Surds |

AQA AS/A2 Maths 2017 | Pure Maths | Algebra | Surds |

CCEA A-Level (NI) | C1 | Algebra and Functions | Surds |

B Algebra and Functions | B2 Surds | Surds | |

Edexcel A-Level (UK - Pre-2017) | C1 | Algebra and Functions | Surds |

Edexcel AS Maths 2017 | Pure Maths | Algebraic Expressions | Surds |

Edexcel AS/A2 Maths 2017 | Pure Maths | Algebraic Expressions | Surds |

I.B. Higher Level | 2 | Algebra and Functions | Surds |

I.B. Standard Level | 1 | Algebra and Functions | Surds |

Methods (UK) | M1 | Algebra and Functions | Surds |

OCR A-Level (UK - Pre-2017) | C1 | Algebra and Functions | Surds |

OCR AS Maths 2017 | Pure Maths | Indices and Surds | Surds |

OCR MEI AS Maths 2017 | Pure Maths | Surds and Indices | Surds |

OCR-MEI A-Level (UK - Pre-2017) | C1 | Algebra and Functions | Surds |

Pre-U A-Level (UK) | 1 | Algebra and Functions | Surds |

Universal (all site questions) | A | Algebra and Functions | Surds |

WJEC A-Level (Wales) | C1 | Algebra and Functions | Surds |