Rationalise the denominator of the following fraction: 1/(√2 + 1)

We start with 1/(√2 + 1) 

Normally with rationalising surd denominators we multiply the top and bottom of the fraction by the denominator. But this time we have a surd ADDED by a rational number. 

In this case we multiply the top and bottom by the denominator with the connecting + or - sign REVERSED ie by:(√2 - 1)

So we get

(√2 - 1) / (√2 + 1)(√2 - 1) 

The bottom is multiplied out like a quadratic... a special type of quadratic [(a - b)(a + b)]. A handy but not vital rule to remember is:

(a - b)(a + b) = a2 - b2 

So back to our fraction, we get

(√2 - 1) / (2 - √2 + √2 - 1)

= (√2 - 1) / (2 - 1)   

= (√2 - 1) / 1

= √2 - 1 --> our final answer!

If you would like more examples, as usual BBC Bitesize is good at walking through the solutions to a number of types of questions on surds: 

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