Fractions like \displaystyle\frac{3}{5} or \frac{ }{ }\displaystyle\frac{x^2 - 2x}{17} have rational
denominators. In other words the number on the bottom of the fraction is a
rational number. The fraction \displaystyle\frac{5}{\sqrt{17} } has an irrational
denominator. This kind of fraction can be converted to a fraction with a
rational denominator. The process of doing this is called "rationalising the
denominator" and consists of multiplying the fraction by a suitable fraction of
the form \displaystyle\frac{a}{a}, where
a is chosen specially. Note that, for any surd
(x+\sqrt{y})(x-\sqrt{y}) = x^2-y

## Summary/Background

"Rationalising the denominator" means changing a fraction so that
the denominator (the term on the bottom) does not have a surd in it. The method
consists of multiplying the fraction by another fraction that actually equals
1. In every case above, the first step is to multiply by a fraction of unit
value.

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

### denominator

the integer on the bottom of a fraction

### fraction

A ratio of two numbers or polynomials.

### rational number

a number that can be expressed as a fraction

### 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 |