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1.8 H) Rationalising Denominators – Part 2
1.8 H) Rationalising Denominators – Part 2
Whenever we give an answer as a fraction, we should make sure that there are no surds in the denominator. The process of getting rid of any surds that are in a denominator is known as rationalising the denominator. The previous section was looking at fractions where the denominator was a single surd (e.g. the denominator was √4 or 2√5). In this section, we are going to be looking at rationalising denominators where there is an ordinary term (number or unknown) and a surd. For example, we are going to be looking at rationalising fraction where the denominator is something like 5 + √2 or √7 – 4.
We are able to rationalise the denominator by multiplying the numerator and the denominator of fraction by the denominator with the opposite sign in. For example, if the denominator of a fraction was 5 + √2, we would multiply the numerator and the denominator by 5 – √2. If the denominator was √7 – 4, we would multiply the numerator and the denominator of the fraction by √7 + 4.
This technique is known as the difference of two squares and the whole point of it is to get rid of the surd in the denominator.
Let’s multiply out the examples of the two brackets that are given in the paragraph above.
We are able to rationalise the denominator by multiplying the numerator and the denominator of fraction by the denominator with the opposite sign in. For example, if the denominator of a fraction was 5 + √2, we would multiply the numerator and the denominator by 5 – √2. If the denominator was √7 – 4, we would multiply the numerator and the denominator of the fraction by √7 + 4.
This technique is known as the difference of two squares and the whole point of it is to get rid of the surd in the denominator.
Let’s multiply out the examples of the two brackets that are given in the paragraph above.
I am going to multiply these brackets out by using the FOIL technique (First, Outside, Inside and Last).
The next step is to simplify the expression. We can simplify the expression by collecting the numbers and the surds. We can combine the numbers together to get 23 (25 – 2). The two surds that we can combine are -5√2 and 5√2. These two terms will cancel one another out. This means that the expression simplified is just 23.
This gives us 23. Therefore, if the denominator of our fraction was 5 + √2 and we multiplied by 5 – √2, the denominator of the fraction would no longer contain a surd, which is the whole aim of rationalising the denominator of a fraction.
The rule for multiplying brackets that are the difference of two squares is:
The rule for multiplying brackets that are the difference of two squares is:
Let’s now look at the second example.
Example 1
Rationalise the denominator for the fraction below.
Rationalise the denominator for the fraction below.
In order the rationalise the denominator, we are going to be multiplying by the numerator and the denominator of the fraction by the denominator with the opposing sign in the middle. The denominator of the fraction is 4 + √2, which means that we are going to be multiplying the numerator and the denominator by 4 – √2.
We obtain the numerator by multiplying each of the terms in the bracket by 6. The two brackets for the denominator are the difference of two squares and this means that when we multiply it out, we will have the first term squared (42) minus the square of the second term ((√2)2).
The final step is to check that the fraction is given in its simplest form. We do this by checking that all of the surds are given in their simplest form and that there are no common factors in the numerator and the denominator. It is best to check whether the surds can be simplified first, and then check whether there are any common factors in the numerator and the denominator.
The only surd that we have in the fraction is √2 and there are no square factors in √2, which means that the surd is in its simplest form.
We now need to check that there are no common factors in the numerator and the denominator. We are looking for common factors in 24, -6 and 14 (for the surd, we are only looking for common factors in the number outside the surd; treat the surd as an unknown). All of these numbers have a common factor of 2. Before we divide by 2, I am going to factorise 2 out of the numerator of the fraction.
The only surd that we have in the fraction is √2 and there are no square factors in √2, which means that the surd is in its simplest form.
We now need to check that there are no common factors in the numerator and the denominator. We are looking for common factors in 24, -6 and 14 (for the surd, we are only looking for common factors in the number outside the surd; treat the surd as an unknown). All of these numbers have a common factor of 2. Before we divide by 2, I am going to factorise 2 out of the numerator of the fraction.
The next step is to divide both the numerator and the denominator by 2. When we divide the numerator by 2, the two on the outside of the bracket goes (this is the reason why I factorised 2 out of the numerator). When we divide the denominator by 2, we get 7.
Therefore, the final answer is: