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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

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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:
  • \sqrt{ab} = \sqrt{a}\sqrt{b}
  • \displaystyle \sqrt{\frac{a}{b} } =\frac{ \sqrt{a} }{\sqrt{b} }
Be aware also of these common mistakes when a and b are both positive:
  • \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}
  • \sqrt{a - b} \ne \sqrt{a} - \sqrt{b}

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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.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopic
AP Calculus AB (USA)1Algebra and FunctionsSurds
AP Calculus BC (USA)1Algebra and FunctionsSurds
AQA A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
AQA AS Maths 2017Pure MathsAlgebraSurds
AQA AS/A2 Maths 2017Pure MathsAlgebraSurds
CCEA A-Level (NI)C1Algebra and FunctionsSurds
B Algebra and FunctionsB2 SurdsSurds
Edexcel A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
Edexcel AS Maths 2017Pure MathsAlgebraic ExpressionsSurds
Edexcel AS/A2 Maths 2017Pure MathsAlgebraic ExpressionsSurds
I.B. Higher Level2Algebra and FunctionsSurds
I.B. Standard Level1Algebra and FunctionsSurds
Methods (UK)M1Algebra and FunctionsSurds
OCR A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
OCR AS Maths 2017Pure MathsIndices and SurdsSurds
OCR MEI AS Maths 2017Pure MathsSurds and IndicesSurds
OCR-MEI A-Level (UK - Pre-2017)C1Algebra and FunctionsSurds
Pre-U A-Level (UK)1Algebra and FunctionsSurds
Universal (all site questions)AAlgebra and FunctionsSurds
WJEC A-Level (Wales)C1Algebra and FunctionsSurds