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
Note that in the first step, I
pulled out 2 as the common factor of the denominator, so that I get a simpler
surd to work with. Always try to work
with simpler expressions. This not only
shortens your working, it reduces your chances of making a careless mistake.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 rationalising the denominator work? This is because on the bottom (denominator) we have a difference-of-squares expression of the form
(a + b)(a – b) which is equal to a2 – b2.
Since squaring “gets rid” of square roots, a2 and b2 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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