Surds and Rationalising Denominators
Neil Trivedi
Teacher
Contents
Surds and Rationalising Denominators
Sometimes, we are asked to rationalise the denominator of an expression such as .
Rationalising the denominator means removing the surd, which is irrational, from the denominator. When there is a single surd in the denominator, we do this by multiplying both the numerator and denominator by that same surd.
This works because , which is a rational number.
For example, consider .
To remove the surd from the denominator, we multiply both the numerator and the denominator by .
The denominator then becomes rational because
So,
Note: We can do this because we are finding an equivalent fraction. By multiplying the numerator and denominator by , we are just multiplying by so the fraction hasn't changed in value.
Example 1:
Rationalise the following:
a)
Single Step: Multiply the numerator and denominator by the surd in the denominator.
b)
Single Step: Multiply the numerator and denominator by the surd in the denominator.
We then simplify the fraction by dividing the numerator and denominator by .
Note: when we divided the numerator and denominator by , the in the root is protected so it does not simplify. This is similar to how .
c)
Single Step: Multiply the numerator and denominator by the surd in the denominator.
Recall that . We use this to simplify the numerator.
We can simplify because , which is a square number, is a factor of and is the largest square number factor. So, we write as .
Splitting the root in the numerator,
Since ,
We can simplify the fraction on the right-hand side by dividing the numerator and denominator by .
Therefore,
Rationalising Denominators with more than one term
When the denominator has more than one term, we rationalise it by using the difference of two squares principle.
Recall that .
The surd expression that we multiply both the numerator and denominator by to rationalise the denominator is known as the conjugate.
Example 2:
Simplify by rationalising:
a)
Single Step: Multiply the numerator and denominator by the conjugate of the denominator and simplify.
Expanding the denominator and simplifying,
Note: Most students’ first instinct is to multiply the numerator and denominator by since that is what we were doing previously. However, if we multiply by , we get
which hasn’t removed the root from the denominator.
b)
Single Step: Multiply the numerator and denominator by the conjugate of the denominator and simplify.
Expanding the numerator and denominator and simplifying,
We then simplify the fraction by dividing the numerator and denominator by .
Note: when expanding the denominator using the difference of two squares, the middle terms will always cancel so you only really need to show the first and last terms which, in this case, is .
c)
Single Step: Multiply the numerator and denominator by the conjugate of the denominator and simplify.
Expanding the numerator and denominator and simplifying,
We then simplify the fraction by dividing the numerator and denominator by .
We can simplify because , which is a square number, is a factor of and is the largest square number factor. So, we write as .
Since ,
Therefore,
Example 3:
The area of this rectangle is m. Find the length of the rectangle labelled , giving your answer in the form where and are constants to be found.
Step 1: Form an equation to solve for .
We know that the area of a rectangle is found by multiplying the width and the length. So,
We isolate by dividing both sides by .
Now, to get in the form , we need to rationalise the denominator.
Step 2: Multiply the numerator and denominator by the conjugate of the denominator and simplify.
Expanding and simplifying the numerator and denominator,
We then simplify the fraction by dividing the numerator and denominator by .
Therefore,
Hence, and .
Challenging Questions