# Surds

## What are surds?

Before we can try to solve surds, we first need to know what they are. Surds are square roots that cannot be reduced to  give us a whole number. They are irrational numbers. However, we can work with them by simplifying them or rationalising the denominator of the surd. Simplifying Surds First, we will try to simplify some surds. The concept of simplifying a surd is quite easy. However, you will have to think out of the box a bit. On the site, there are three examples of simplification. In the first one, we have converted the sqrt of 20 into a multiplication sum: sqrt of 5 x sqrt of 4. Since we know that the square root of 4 is 2, we can simplify this expression by writing it as: 2 x sqrt 5. In the second expression, we have to multiply two surds together. To gives us the square root of 81. Since the square root of 81 is equal to 9, the answer to this sum is 9. In the last example, we have divided one square root by another. This is possible because the sqrt of 16 over the sqrt of 8 is generally seen as the sqrt of (16/8). Solving the fraction then allows us to give our answer as sqrt  of 2. These example should have given you a general idea of how simplifying surds works. You basically have to divide and multiply to see if you can simplify the surd or even solve part of it as in example 1. Examples 2 and 3 were just there to show you how to multiply and divide surds to simplify them. Underneath you will find some more examples. Rationalising Denominators of Surds In mathematics, we do not like to have surds as denominators of fractions. Consequently, we rationalise them to make the fraction more sophisticated. Look at the example on the right. We start off with the fraction sqrt 5 / sqrt 8. To rationalise the denominator in this example, we just need to multiply it by sqrt 8 / sqrt 8 which is equivalent to 1. Sqrt 5 x sqrt 8 gives us sqrt 40 and on the bottom, when we times sqrt 8 by sqrt 8 we get 8 because any square root times by itself is equal to the number inside the square root. The rationalised fraction is sqrt 40 / 8. There are a few other situations in which the method is the same. Look at the examples underneath. Presentation Discover More! I hope you have understood everything on this page about surds. Make sure to check out some other topics on this website to improve your maths skills in other areas too by visiting our library page. Please share this page if you like it or found it helpful!
Example 1 In this example, we split the square root into two other square roots (9 and 5). We can reduce sqrt 9 to 3 and are so able to give a simplified answer.
Example 2 In this example, we multiplied two sqrts together which gave us the sqrt of 144 that we can reduce to 12.
Example 3 In this example, we divided the two sqrts by each other to give us the sqrt of 4.
Example 1 In this example, we have a similar situation to the one in the example above. We apply the same method to rationalise the denominator.
Example 2 In this example, we have a number as the numerator and a number and a surd as a denominator. To rationalise the denominator, we just times the fraction by sqrt 2 / sqrt 2 (=1) which gives us 2 times 2 which is equal to 4 as the new denominator and then we just need to solve the top by multiplying 5 and sqrt 2 which is 5 sqrt 2.
Example 3 In this last example, we have a number multiplied by a surd on the top and bottom of the fraction. The procedure is still, the same, we times the fraction by sqrt 6 / sqrt 6 which gives us 7 sqrt 48 as the numerator and 30 as the denominator of the rationalised fraction.
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Chapter 7.3:  Learning Outcomes Students will know what surds are! Students will be able to simplify surds! Students will be able to rationalise denominators in surd form!

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