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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



"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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the integer on the bottom of a fraction


A ratio of two numbers or polynomials.

square root

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


A number containing one or more irrational square roots.