**Introduction**

Expressions
with surds in their denominators are cumbersome. The good news is that we can make the
denominators into

*rational numbers*, which are nicer.**Rational numbers**those that can be expressed as a*ratio*of integers i.e. they are (proper or improper) fractions or can be converted to fractions. Whole numbers are also part of rational numbers because you can always put them upon a denominator of 1; e.g. 2 =^{2}/_{1}, so 2 is a rational number.
The standard trick for simplifying expressions
with surds in their denominators is to

**rationalise the denominator**by mutiplying the numerator and the denominator with its conjugate surd. For example, the**conjugate surd**of Ö5 + Ö2 is Ö5 – Ö2. Just change the + to – or the – to +. Let us see how the magic works.**Solution**

**Remarks**

In mathematics, “rationalising” does not mean you give some reason or excuse for something that you know you have done wrong. It means “make it into a rational number”. Why does

**work? This is because on the bottom (denominator) we have a difference-of-squares expression of the form**

*rationalising the denominator*(

*a*+

*b*)(

*a*–

*b*) which is equal to

*a*

^{2}–

*b*

^{2}.

Since squaring “gets rid” of square roots,

*a*

^{2}and

*b*

^{2}will give you rational numbers (whole numbers or fractions), you will end up with a nice number downstairs (on the denominator). Pupils should make sure they have this technique in their repertoire of skills.

**Suitable Levels**

*****GCE ‘O’ Level Additional Mathematics

* revision for GCE ‘A’ Level H2 Mathematics

* revision for IB Mathematics HL / SL

* other syllabuses that involve surds

* precocious kids who always want to learn more

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