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Rationalizing the Denominator: Dealing With Surds/Radicals in Fractions

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.

Fractions With irrational denominators

Fractions With irrational denominators

What Is an Irrational Number?

Before looking at irrational numbers, it helps to understand what a rational number is. A rational number is defined as any number that can be written as a fraction p/q where p and q are both integers (whole numbers). 7 is a rational number, as it can be written as 7/1. 0.34 is also a rational number, as it can be written as 34/100 or simplified to 17/50.

A further helpful identifier for rational numbers is that, when written in decimal form, the number either comes to an end such as 0.5 or 0.3958, or it recurs such as 0.3333... or 0.17171717...

An irrational number is any number that is not rational, and so cannot be written as a fraction of whole numbers. When written in decimal form, their decimals continue forever without repetition or pattern. π and e are two very famous examples of irrational numbers, but interestingly, so is any square root of a whole number that isn't itself a square number. For example, √2 and √3 are irrational, √4 = 2, and is therefore rational. Numbers made up of irrational roots are called surds (or radicals in the US), and it is these we are going to look at here.

What Is Rationalising the Denominator?

When dealing with fractions involving surds, it is usually regarded as best practice to have a rational number on the bottom (the denominator) and leave any irrational numbers to the top (the numerator). We call this rationalizing the denominator. To see the reasoning behind this, take a look at my article 'Why Do We Rationalise the Denominator?'

Dealing With Simple Surd Fractions

To start with, let's look at a simple example; 1/√2.

To rationalize the denominator here, we use the fact that the square root of a number n, multiplied by itself is n. i.e. √2 × √2 = 2.

Multiplying the top and bottom of the fraction by √2 will therefore give us a rational denominator without changing the value of the fraction.


We can now see that 1/√2 = √2 / 2.

These are exactly the same number (equal to 0.7071067812...) but we have now converted it so that the fraction has a rational denominator.

Example 2

Now let's look at a similar example: 3/√12.

We could multiply top and bottom by √12 and this will give us a correct answer, but it is always good practice to see if the root can be simplified first.

12 = 3×4 and 4 is a square number, so √12 = √3 × √4 = 2√3.

We now have 3/√12 = 3 / 2√3.

We can now multiply top and bottom by √3 to rationalize the denominator. √3 × √3 = 3 which cancels with the 3 in the numerator to leave us with √3 / 2.



A Trickier Example

What about fractions of the form 3 / (4 − √5)? We can't just multiply top and bottom by √5 as this will still leave us with a 4√5 in the denominator.

Instead, we use something called the 'conjugate'. The conjugate of 4 − √5 is 4 + √5 (just replace the minus with an add, or vice versa if required). We now multiply top and bottom by this conjugate, making use of the difference of two squares rule to cancel out the roots in the denominator as below. (The difference of two squares rule is that (a + b)(a − b) = a2 − b2).




So, the method used to rationalize the denominator depends upon what format the original fraction is in.

  1. If the denominator is just a square root (or multiple of a square root), simply multiply top and bottom by this square root and then simplify as required.
  2. If the denominator is of the form a ± b√c, then multiply by its conjugate formed by switching the minus for a plus or vice versa. Again, simplify afterward if required.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 David