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1.8 G) Rationalising Denominators – Part 1
1.8 G) Rationalising Denominators – Part 1
We should never give a fraction where the denominator contains a surd. This is because it is often easier to use fractions that do not have surds in their denominators. The fraction below has a surd in its denominator.
The process of getting rid of a surd when it is on the denominator is known as rationalising the denominator.
Single Surds as Denominators
In this section, we are going to be looking at rationalising the denominator of a fraction that has single term containing a surd. In order to rationalise this type of fraction, we multiply the numerator and the denominator of the fraction by the surd that is on denominator. We need to make sure that we multiply both the numerator and the denominator by the surd because we want to create an equivalent fraction and when we create equivalent fractions, we need to multiply both the numerator and the denominator by the same thing.
In this section, we are going to be looking at rationalising the denominator of a fraction that has single term containing a surd. In order to rationalise this type of fraction, we multiply the numerator and the denominator of the fraction by the surd that is on denominator. We need to make sure that we multiply both the numerator and the denominator by the surd because we want to create an equivalent fraction and when we create equivalent fractions, we need to multiply both the numerator and the denominator by the same thing.
Example 1
Rationalise the denominator for the fraction below:
Rationalise the denominator for the fraction below:
In order to rationalise this fraction, we are going to multiply the numerator and the denominator of the fraction by the surd in the denominator of the fraction; we will be multiplying the numerator and the denominator by √2.
Currently, the numerator of the fraction is 6 and we are multiplying it by √2. This is going to result in the new numerator being 6√2.
Currently, the denominator of the fraction is √2 and we are multiplying it by √2. When we multiply a surd by the same surd, the answer is the number that is inside the surds (the rule is: √a x √a = a). Therefore, √2 multiplied by √2 is 2. This means that the denominator of the fraction is 2. We have now successfully got rid of the surd from the denominator of the fraction.
We now need to check whether our answer can be simplified. There are two different simplifications that we need to check for.
The first one is to check whether any of the surds in the fraction can be simplified. We check whether surds can be simplified by looking for square factors in the numbers inside the surds. If there are any square factors, we can take the square factors out of the surd. There are no square factors in 2, which means that the surd cannot be simplified.
The second simplification step is to check whether the fraction can be simplified and we do this by looking for any common factors that go into the number outside the surd in the numerator and the denominator; we are looking for any common factors between 6 and 2. 2 is a factor of both 6 and 2, which means that we can divide the numerator and the denominator by 2.
The first one is to check whether any of the surds in the fraction can be simplified. We check whether surds can be simplified by looking for square factors in the numbers inside the surds. If there are any square factors, we can take the square factors out of the surd. There are no square factors in 2, which means that the surd cannot be simplified.
The second simplification step is to check whether the fraction can be simplified and we do this by looking for any common factors that go into the number outside the surd in the numerator and the denominator; we are looking for any common factors between 6 and 2. 2 is a factor of both 6 and 2, which means that we can divide the numerator and the denominator by 2.
Currently the numerator is 6√2 and we are dividing it by 2. The best way to think about this calculation is to imagine that the surd is an unknown (like x), which would mean that we have 6x. We are dividing 6x by 2, which means that we now have 3x. We now replace the x with √2, and this means that our 6√2 ÷ 2 is 3√2.
The current denominator is 2 and when we divide it by 2, we get 1.
Anything over 1 is the just the numerator, which means that the final answer to this question is:
Example 2
Rationalise the denominator for the fraction below:
Rationalise the denominator for the fraction below:
To rationalise the denominator, we are going to be multiplying the numerator and the denominator of the fraction by the surd that on the denominator; we multiply the numerator and the denominator by √10.
Currently the numerator of the fraction is √6 and we are multiplying it by √10. When we are multiplying surds, we multiply the numbers that are inside the surds. The rule is: √a x √b = √ab. Therefore, when we multiply √6 by √10, we get: √6x10 = √60.
Currently the denominator of the fraction is √10 and we are multiplying it by √10. Whenever we multiply a surd by the same surd, the answer is the number inside the surd. The rule is √a x √a = a. Therefore, √10 multiplied by √10 is 10.
Currently the denominator of the fraction is √10 and we are multiplying it by √10. Whenever we multiply a surd by the same surd, the answer is the number inside the surd. The rule is √a x √a = a. Therefore, √10 multiplied by √10 is 10.
We now need to simplify to check whether our answer can be simplified. We need to check whether the surd can be simplified and whether the fraction can be simplified.
To simplify the surd, we are looking for square factors that go into 60. A square factor that goes into 60 is 4. Therefore, we can rewrite the numerator as:
To simplify the surd, we are looking for square factors that go into 60. A square factor that goes into 60 is 4. Therefore, we can rewrite the numerator as:
Our fraction now becomes:
The second step is to simplify the fraction, which we do by dividing by any common factors between the number outside the surd in the numerator and the denominator; we are looking for common factors between 2 and 10. There is a common factor of 2, so we divide the numerator and the denominator by 2.
This means that the fraction becomes:
Example 3
Rationalise the denominator for the fraction below:
Rationalise the denominator for the fraction below:
Whenever we have a denominator that is just one term, we rationalise the denominator by multiplying the numerator and the denominator of the fraction by just the surd that is in the denominator, which in this case is √3 (you can multiply the numerator and the denominator by 7√3, but you will just need to simplify your answer at the end, which is a little more time consuming).
Currently the numerator of the fraction is 6√5 and we want to multiply it by √3. The easiest way to multiply these surds is to imagine the second surd as 1√3. We are now multiplying 6√5 by 1√3. This is going to give us an answer in the form of a√b. We obtain the value for a by multiplying the numbers on the outside of the surds; we multiply 6 by 1, which means that a is 6. We obtain the value for b by multiplying the numbers that are on the inside of the surds; we multiply 5 by 3, which means that b is 15. Therefore, 6√5 multiplied by √3 is 6√15.
Currently the denominator of the fraction is 7√3 and we are multiplying it by √3. The calculation that we are undertaking is: 7√3 x √3. We can modify this calculation by writing the 7√3 as 7 x √3. This means that the calculation becomes: 7 x √3 x √3. Whenever we multiply a surd by the same surd, we obtain the number that is inside the surd (the rule is: √a x √a = a) Therefore, the √3 multiplied by √3 will become 3. This means that our calculation becomes: 7 x 3, which is 21.
The next step is to check whether our fraction can be simplified.
Let’s first check whether the surd is in its simplest form. There are no square factors in 15, which means that the surd is already in its simplest form.
The second check is to simplify the fraction, which we do by dividing by any common factors between the number outside the surd in the numerator and the denominator; we are looking for common factors between 6 and 21. There is a common factor of 3, so we divide both the numerator and the denominator by 3.
Let’s first check whether the surd is in its simplest form. There are no square factors in 15, which means that the surd is already in its simplest form.
The second check is to simplify the fraction, which we do by dividing by any common factors between the number outside the surd in the numerator and the denominator; we are looking for common factors between 6 and 21. There is a common factor of 3, so we divide both the numerator and the denominator by 3.
This means that the final answer is: