How do you 'rationalise the denominator'?

There are two approaches to rationalising a denominator depending on how the denominator appears.

First we deal with the simpler case where the surd term is the only term in the denominator.

For example: 6/√3, 4/(3√7) or (5+√2)/√3

The trick of 'multiplying by one' can be used on each of these fractions to give an alternate expression that does not contain a surd in the denominator.

Let's consider the fraction 6/√3

If we multiply this fraction by √3/√3 we do not change the value of the fraction because √3/√3=1

Let's go ahead and do the multiplication:

The numerator is 6* √3 which is most simply expressed as 6√3

The denominator is √3*√3 which by the definition of the square root function is equal to 3

So putting the numerator and denominator together leaves us with the fraction (6√3)/3

Some basic cancellation reduces this to the simpler form of 2√3

So 6/√3=2√3

Next comes the more complicated case where the surd term is not the only term in the denominator.

For example: 1/(1-√3), (1+√2)/(3+√2) or 6/(2√5+√3)

The trick of 'multiplying by one' can again be used but it is less clear what 'one' to use.

To find our 'one' we must recall the difference of two squares formula: a²-b²=(a+b)(a-b)

Let's consider 1/(1-√3)

From the difference of two squares formula we can see that if we multiply the denominator by 1+√3 then we are left with 1²-(√3)²=1-3=-2

From this we can infer that it is necessary to multiply the fraction by (1+√3)/(1+√3)

Let's go ahead and do the multiplication:

The numerator is 1*(1+√3) which is equal to 1+√3

The denominator we have already calculated is -2

So putting the numerator and the denominator together leaves us with (1+√3)/-2

So 1/(1-√3)=(1+√3)/-2

A useful reason for rationalising the denominator is that it helps when thinking about what value a fraction really represents. For example, when considering the fraction 1/√2=√2/2, it is hard to imagine 1 divided into √2 pieces as √2 is an irrational number. It makes more sense however to imagine √2 divided into 2 pieces as that is just a simple halving and it doesn't matter too much that √2 is irrational.